-- Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
lemma copy {P' : MorphismProperty Scheme.{u}}
    {Q' : ∀ {R S : Type u} [CommRing R] [CommRing S], (R →+* S) → Prop}
    (e : P = P') (e' : ∀ {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S), Q f ↔ Q' f) :
    HasRingHomProperty P' Q'

Proof state:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
heq : Q = Q'
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q'

Tactic:
rw [← heq]

Result:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
heq : Q = Q'
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q

Proof state:
case h.h.h.h.h.a
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
R S : Type u
x✝¹ : CommRing R
x✝ : CommRing S
f : R →+* S
⊢ Q f ↔ Q' f

Tactic:
exact (e' f)

Result:
no goals

Proof state:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
heq : Q = Q'
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q

Tactic:
infer_instance

Result:
no goals

Proof state:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
⊢ Q = Q'

Tactic:
ext R S _ _ f

Result:
case h.h.h.h.h.a
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
R S : Type u
x✝¹ : CommRing R
x✝ : CommRing S
f : R →+* S
⊢ Q f ↔ Q' f

Proof state:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q'

Tactic:
have heq : @Q = @Q' := by
  ext R S _ _ f
  exact (e' f)

Result:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
heq : Q = Q'
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q'

Proof state:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
P' : MorphismProperty Scheme
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e : P = P'
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
⊢ HasRingHomProperty P' fun {R S} [CommRing R] [CommRing S] => Q'

Tactic:
subst e

Result:
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
e' : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S), Q f ↔ Q' f
⊢ HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] => Q'