Theorem vtocl2g 2794

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)

Hypotheses

Ref	Expression
vtocl2g.1	|- ( x = A -> ( ph <-> ps ) )
vtocl2g.2	|- ( y = B -> ( ps <-> ch ) )
vtocl2g.3	|- ph

Assertion

Ref	Expression
vtocl2g	|- ( ( A e. V /\ B e. W ) -> ch )

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

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

Proof of Theorem vtocl2g

Step	Hyp	Ref	Expression
1		nfcv 2312	. 2 |- F/_ x A
2		nfcv 2312	. 2 |- F/_ y A
3		nfcv 2312	. 2 |- F/_ y B
4		nfv 1521	. 2 |- F/ x ps
5		nfv 1521	. 2 |- F/ y ch
6		vtocl2g.1	. 2 |- ( x = A -> ( ph <-> ps ) )
7		vtocl2g.2	. 2 |- ( y = B -> ( ps <-> ch ) )
8		vtocl2g.3	. 2 |- ph
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gf 2792	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  uniprg  3811  intprg  3864  opthg  4223  opelopabsb  4245  unexb  4427  vtoclr  4659  elimasng  4979  cnvsng  5096  funopg  5232  f1osng  5483  fsng  5669  fvsng  5692  op1stg  6129  op2ndg  6130  xpsneng  6800  xpcomeng  6806  bdunexb  13955  bj-unexg  13956