Theorem vtocl2ga 2688

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl2ga.1	|- ( x = A -> ( ph <-> ps ) )
vtocl2ga.2	|- ( y = B -> ( ps <-> ch ) )
vtocl2ga.3	|- ( ( x e. C /\ y e. D ) -> ph )

Assertion

Ref	Expression
vtocl2ga	|- ( ( A e. C /\ B e. D ) -> ch )

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

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

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 |- F/_ x A
2		nfcv 2229	. 2 |- F/_ y A
3		nfcv 2229	. 2 |- F/_ y B
4		nfv 1467	. 2 |- F/ x ps
5		nfv 1467	. 2 |- F/ y ch
6		vtocl2ga.1	. 2 |- ( x = A -> ( ph <-> ps ) )
7		vtocl2ga.2	. 2 |- ( y = B -> ( ps <-> ch ) )
8		vtocl2ga.3	. 2 |- ( ( x e. C /\ y e. D ) -> ph )
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gaf 2687	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622

This theorem is referenced by:  caovcan  5823  genipv  7129  fsumrelem  10926