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Metamath Proof Explorer

Theorem vtocl2gaf

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013) (Proof shortened by Wolf Lammen, 31-May-2025)

Ref Expression
Hypotheses vtocl2gaf.a 
|- F/_ x A
vtocl2gaf.b 
|- F/_ y A
vtocl2gaf.c 
|- F/_ y B
vtocl2gaf.1 
|- F/ x ps
vtocl2gaf.2 
|- F/ y ch
vtocl2gaf.3 
|- ( x = A -> ( ph <-> ps ) )
vtocl2gaf.4 
|- ( y = B -> ( ps <-> ch ) )
vtocl2gaf.5 
|- ( ( x e. C /\ y e. D ) -> ph )
Assertion vtocl2gaf 
|- ( ( A e. C /\ B e. D ) -> ch )

Proof

Step Hyp Ref Expression
1 vtocl2gaf.a 
 |-  F/_ x A
2 vtocl2gaf.b 
 |-  F/_ y A
3 vtocl2gaf.c 
 |-  F/_ y B
4 vtocl2gaf.1 
 |-  F/ x ps
5 vtocl2gaf.2 
 |-  F/ y ch
6 vtocl2gaf.3 
 |-  ( x = A -> ( ph <-> ps ) )
7 vtocl2gaf.4 
 |-  ( y = B -> ( ps <-> ch ) )
8 vtocl2gaf.5 
 |-  ( ( x e. C /\ y e. D ) -> ph )
9 2 nfel1 
 |-  F/ y A e. C
10 9 5 nfim 
 |-  F/ y ( A e. C -> ch )
11 7 imbi2d 
 |-  ( y = B -> ( ( A e. C -> ps ) <-> ( A e. C -> ch ) ) )
12 nfv 
 |-  F/ x y e. D
13 12 4 nfim 
 |-  F/ x ( y e. D -> ps )
14 6 imbi2d 
 |-  ( x = A -> ( ( y e. D -> ph ) <-> ( y e. D -> ps ) ) )
15 8 ex 
 |-  ( x e. C -> ( y e. D -> ph ) )
16 1 13 14 15 vtoclgaf 
 |-  ( A e. C -> ( y e. D -> ps ) )
17 16 com12 
 |-  ( y e. D -> ( A e. C -> ps ) )
18 3 10 11 17 vtoclgaf 
 |-  ( B e. D -> ( A e. C -> ch ) )
19 18 impcom 
 |-  ( ( A e. C /\ B e. D ) -> ch )