Theorem: LO.FirstOrder.Structure.Eq.eqv_refl (Logic/FirstOrder/Basic/Eq.lean)

State before:
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
⊢ eqv L a a
Tactic:
have : M ⊧ₘ “∀ x, x = x” := H.realize (Theory.eqAxiom.refl (L := L))
State after:
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : M ⊧ₘ (“∀' #0 = #0”)
⊢ eqv L a a

State before:
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : M ⊧ₘ (“∀' #0 = #0”)
⊢ eqv L a a
Tactic:
simp [models_def] at this
State after:
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a

State before:
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a
Tactic:
exact this a
State after:
no goals