Theorem vtocl2g 2745

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 2279	. 2 |- F/_ x A
2		nfcv 2279	. 2 |- F/_ y A
3		nfcv 2279	. 2 |- F/_ y B
4		nfv 1508	. 2 |- F/ x ps
5		nfv 1508	. 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 2743	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  uniprg  3746  intprg  3799  opthg  4155  opelopabsb  4177  unexb  4358  vtoclr  4582  elimasng  4902  cnvsng  5019  funopg  5152  f1osng  5401  fsng  5586  fvsng  5609  op1stg  6041  op2ndg  6042  xpsneng  6709  xpcomeng  6715  bdunexb  13107  bj-unexg  13108