Theorem vtocl2gf 2748

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 2697	. 2 |- ( A e. V -> A e. _V )
2		vtocl2gf.3	. . 3 |- F/_ y B
3		vtocl2gf.2	. . . . 5 |- F/_ y A
4	3	nfel1 2292	. . . 4 |- F/ y A e. _V
5		vtocl2gf.5	. . . 4 |- F/ y ch
6	4, 5	nfim 1551	. . 3 |- F/ y ( A e. _V -> ch )
7		vtocl2gf.7	. . . 4 |- ( y = B -> ( ps <-> ch ) )
8	7	imbi2d 229	. . 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 2744	. . 3 |- ( A e. _V -> ps )
14	2, 6, 8, 13	vtoclgf 2744	. 2 |- ( B e. W -> ( A e. _V -> ch ) )
15	1, 14	mpan9 279	1 |- ( ( A e. V /\ B e. W ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2268   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  vtocl3gf  2749  vtocl2g  2750  vtocl2gaf  2753