Theorem vtocl2g 2837

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 2348	. 2 |- F/_ x A
2		nfcv 2348	. 2 |- F/_ y A
3		nfcv 2348	. 2 |- F/_ y B
4		nfv 1551	. 2 |- F/ x ps
5		nfv 1551	. 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 2835	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  uniprg  3865  intprg  3918  opthg  4283  opelopabsb  4307  unexb  4490  vtoclr  4724  elimasng  5051  cnvsng  5169  funopg  5306  f1osng  5565  fsng  5755  fvsng  5782  op1stg  6238  op2ndg  6239  xpsneng  6919  xpcomeng  6925  mhmlem  13483  bdunexb  15893  bj-unexg  15894