Theorem vtocl2gf 2660

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)

Hypotheses

Ref	Expression
vtocl2gf.1	|- F/_ x A
vtocl2gf.2	|- F/_ y A
vtocl2gf.3	|- F/_ y B
vtocl2gf.4	|- F/ x ps
vtocl2gf.5	|- F/ y ch
vtocl2gf.6	|- ( x = A -> ( ph <-> ps ) )
vtocl2gf.7	|- ( y = B -> ( ps <-> ch ) )
vtocl2gf.8	|- ph

Assertion

Ref	Expression
vtocl2gf	|- ( ( A e. V /\ B e. W ) -> ch )

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. V -> A e. _V )
2		vtocl2gf.3	. . 3 |- F/_ y B
3		vtocl2gf.2	. . . . 5 |- F/_ y A
4	3	nfel1 2229	. . . 4 |- F/ y A e. _V
5		vtocl2gf.5	. . . 4 |- F/ y ch
6	4, 5	nfim 1504	. . 3 |- F/ y ( A e. _V -> ch )
7		vtocl2gf.7	. . . 4 |- ( y = B -> ( ps <-> ch ) )
8	7	imbi2d 228	. . 3 |- ( y = B -> ( ( A e. _V -> ps ) <-> ( A e. _V -> ch ) ) )
9		vtocl2gf.1	. . . 4 |- F/_ x A
10		vtocl2gf.4	. . . 4 |- F/ x ps
11		vtocl2gf.6	. . . 4 |- ( x = A -> ( ph <-> ps ) )
12		vtocl2gf.8	. . . 4 |- ph
13	9, 10, 11, 12	vtoclgf 2657	. . 3 |- ( A e. _V -> ps )
14	2, 6, 8, 13	vtoclgf 2657	. 2 |- ( B e. W -> ( A e. _V -> ch ) )
15	1, 14	mpan9 275	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601

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:  vtocl3gf  2661  vtocl2g  2662  vtocl2gaf  2665