Theorem vtocl2ga 2794

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

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

Assertion

Ref	Expression
vtocl2ga	|- ( ( A e. C /\ B e. D ) -> ch )

Distinct variable groups:   x, y, A   y, B   x, C, y   x, D, y   ps, x   ch, y

Allowed substitution hints:   ph( x, y)   ps( y)   ch( x)   B( x)

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		nfcv 2308	. 2 |- F/_ x A
2		nfcv 2308	. 2 |- F/_ y A
3		nfcv 2308	. 2 |- F/_ y B
4		nfv 1516	. 2 |- F/ x ps
5		nfv 1516	. 2 |- F/ y ch
6		vtocl2ga.1	. 2 |- ( x = A -> ( ph <-> ps ) )
7		vtocl2ga.2	. 2 |- ( y = B -> ( ps <-> ch ) )
8		vtocl2ga.3	. 2 |- ( ( x e. C /\ y e. D ) -> ph )
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gaf 2793	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  caovcan  6006  genipv  7450  fsumrelem  11412