Theorem vtocl2gf 2917

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

Hypotheses

Ref	Expression
vtocl2gf.1	|- F/_xA
vtocl2gf.2	|- F/_yA
vtocl2gf.3	|- F/_yB
vtocl2gf.4	|- F/xps
vtocl2gf.5	|- F/ych
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 2868	. 2 |- (A e. V -> A e. _V)
2		vtocl2gf.3	. . 3 |- F/_yB
3		vtocl2gf.2	. . . . 5 |- F/_yA
4	3	nfel1 2500	. . . 4 |- F/y A e. _V
5		vtocl2gf.5	. . . 4 |- F/ych
6	4, 5	nfim 1813	. . 3 |- F/y(A e. _V -> ch)
7		vtocl2gf.7	. . . 4 |- (y = B -> (ps <-> ch))
8	7	imbi2d 307	. . 3 |- (y = B -> ((A e. _V -> ps) <-> (A e. _V -> ch)))
9		vtocl2gf.1	. . . 4 |- F/_xA
10		vtocl2gf.4	. . . 4 |- F/xps
11		vtocl2gf.6	. . . 4 |- (x = A -> (ph <-> ps))
12		vtocl2gf.8	. . . 4 |- ph
13	9, 10, 11, 12	vtoclgf 2914	. . 3 |- (A e. _V -> ps)
14	2, 6, 8, 13	vtoclgf 2914	. 2 |- (B e. W -> (A e. _V -> ch))
15	1, 14	mpan9 455	1 |- ((A e. V /\ B e. W) -> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtocl3gf  2918  vtocl2g  2919  vtocl2gaf  2922