Theorem vtocl2g 3270

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 2764	. 2 |- F/_ x A
2		nfcv 2764	. 2 |- F/_ y A
3		nfcv 2764	. 2 |- F/_ y B
4		nfv 1843	. 2 |- F/ x ps
5		nfv 1843	. 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 3268	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  vtocl4g  3277  ifexg  4157  uniprg  4450  intprg  4511  opthg  4946  opelopabsb  4985  vtoclr  5164  elimasng  5491  cnvsng  5621  funopg  5922  f1osng  6177  fsng  6404  fvsng  6447  fnpr2g  6474  unexb  6958  op1stg  7180  op2ndg  7181  xpsneng  8045  xpcomeng  8052  sbth  8080  unxpdom  8167  fpwwe2lem5  9456  prcdnq  9815  mhmlem  17535  carsgmon  30376  br1steqgOLD  31674  br2ndeqgOLD  31675  brimageg  32034  brdomaing  32042  brrangeg  32043  rankung  32273  mbfresfi  33456  zindbi  37511  2sbc6g  38616  2sbc5g  38617  fmulcl  39813