Theorem: FOL.NV.T_18_8 (FOL/NV/Margaris/Fol.lean)

State before:
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v))
Tactic:
simp only [def_exists_]
State after:
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_)

State before:
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_)
Tactic:
apply IsDeduct.mp_ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))
State after:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    (((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_))

case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))

State before:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    (((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_))
Tactic:
simp only [def_iff_]
State after:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    ((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
      (((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
        ((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_)))

State before:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    ((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
      (((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
        ((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_)))
Tactic:
simp only [def_and_]
State after:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    ((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
            ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
      (((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
          ((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_)

State before:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
    ((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
            ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
      (((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
          ((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_)
Tactic:
SC
State after:
no goals

State before:
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))
Tactic:
apply T_18_6
State after:
case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v

State before:
case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v
Tactic:
exact similar_not P_u P_v u v h1
State after:
no goals