Theorem vtocl2g 2662

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 2219	. 2 |- F/_ x A
2		nfcv 2219	. 2 |- F/_ y A
3		nfcv 2219	. 2 |- F/_ y B
4		nfv 1461	. 2 |- F/ x ps
5		nfv 1461	. 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 2660	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  uniprg  3616  intprg  3669  opthg  3993  opelopabsb  4015  unexb  4195  vtoclr  4406  elimasng  4713  cnvsng  4826  funopg  4954  f1osng  5187  fsng  5357  fvsng  5380  op1stg  5797  op2ndg  5798  xpsneng  6319  xpcomeng  6325  bdunexb  10711  bj-unexg  10712