Theorem vtocl2gf 2840

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 2788	. 2 |- ( A e. V -> A e. _V )
2		vtocl2gf.3	. . 3 |- F/_ y B
3		vtocl2gf.2	. . . . 5 |- F/_ y A
4	3	nfel1 2361	. . . 4 |- F/ y A e. _V
5		vtocl2gf.5	. . . 4 |- F/ y ch
6	4, 5	nfim 1596	. . 3 |- F/ y ( A e. _V -> ch )
7		vtocl2gf.7	. . . 4 |- ( y = B -> ( ps <-> ch ) )
8	7	imbi2d 230	. . 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 2836	. . 3 |- ( A e. _V -> ps )
14	2, 6, 8, 13	vtoclgf 2836	. 2 |- ( B e. W -> ( A e. _V -> ch ) )
15	1, 14	mpan9 281	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1484   e. wcel 2178   F/_wnfc 2337   _Vcvv 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  vtocl3gf  2841  vtocl2g  2842  vtocl2gaf  2845