Theorem vtocl2gf 2863

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 2811	. 2 |- ( A e. V -> A e. _V )
2		vtocl2gf.3	. . 3 |- F/_ y B
3		vtocl2gf.2	. . . . 5 |- F/_ y A
4	3	nfel1 2383	. . . 4 |- F/ y A e. _V
5		vtocl2gf.5	. . . 4 |- F/ y ch
6	4, 5	nfim 1618	. . 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 2859	. . 3 |- ( A e. _V -> ps )
14	2, 6, 8, 13	vtoclgf 2859	. 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 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  vtocl3gf  2864  vtocl2g  2865  vtocl2gaf  2868