Theorem: LO.System.unprovable_imp_trans! (Logic/Logic/HilbertStyle/Basic.lean)

State before:
S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p q r : F
hpq : 𝓢 ⊢! p ⟶ q
⊢ 𝓢 ⊬! p ⟶ r → 𝓢 ⊬! q ⟶ r
Tactic:
contrapose
State after:
S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p q r : F
hpq : 𝓢 ⊢! p ⟶ q
⊢ ¬𝓢 ⊬! q ⟶ r → ¬𝓢 ⊬! p ⟶ r

State before:
S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p q r : F
hpq : 𝓢 ⊢! p ⟶ q
⊢ ¬𝓢 ⊬! q ⟶ r → ¬𝓢 ⊬! p ⟶ r
Tactic:
simp [neg_neg]
State after:
S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p q r : F
hpq : 𝓢 ⊢! p ⟶ q
⊢ 𝓢 ⊢! q ⟶ r → 𝓢 ⊢! p ⟶ r

State before:
S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p q r : F
hpq : 𝓢 ⊢! p ⟶ q
⊢ 𝓢 ⊢! q ⟶ r → 𝓢 ⊢! p ⟶ r
Tactic:
exact imp_trans! hpq
State after:
no goals