Theorem vtocl2ga 2675

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 2223	. 2 |- F/_ x A
2		nfcv 2223	. 2 |- F/_ y A
3		nfcv 2223	. 2 |- F/_ y B
4		nfv 1462	. 2 |- F/ x ps
5		nfv 1462	. 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 2674	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612

This theorem is referenced by:  caovcan  5716  genipv  6813