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Metamath Proof Explorer

Theorem vtocl2ga

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995) Avoid ax-10 and ax-11 . (Revised by GG, 20-Aug-2023) (Proof shortened by Wolf Lammen, 23-Aug-2023)

Ref Expression
Hypotheses vtocl2ga.1 
|- ( x = A -> ( ph <-> ps ) )
vtocl2ga.2 
|- ( y = B -> ( ps <-> ch ) )
vtocl2ga.3 
|- ( ( x e. C /\ y e. D ) -> ph )
Assertion vtocl2ga 
|- ( ( A e. C /\ B e. D ) -> ch )

Proof

Step Hyp Ref Expression
1 vtocl2ga.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 vtocl2ga.2 
 |-  ( y = B -> ( ps <-> ch ) )
3 vtocl2ga.3 
 |-  ( ( x e. C /\ y e. D ) -> ph )
4 2 imbi2d 
 |-  ( y = B -> ( ( A e. C -> ps ) <-> ( A e. C -> ch ) ) )
5 1 imbi2d 
 |-  ( x = A -> ( ( y e. D -> ph ) <-> ( y e. D -> ps ) ) )
6 3 ex 
 |-  ( x e. C -> ( y e. D -> ph ) )
7 5 6 vtoclga 
 |-  ( A e. C -> ( y e. D -> ps ) )
8 7 com12 
 |-  ( y e. D -> ( A e. C -> ps ) )
9 4 8 vtoclga 
 |-  ( B e. D -> ( A e. C -> ch ) )
10 9 impcom 
 |-  ( ( A e. C /\ B e. D ) -> ch )