Theorem vtocl3gf 2917

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtocl3gf.a	|- F/_xA
vtocl3gf.b	|- F/_yA
vtocl3gf.c	|- F/_zA
vtocl3gf.d	|- F/_yB
vtocl3gf.e	|- F/_zB
vtocl3gf.f	|- F/_zC
vtocl3gf.1	|- F/xps
vtocl3gf.2	|- F/ych
vtocl3gf.3	|- F/zth
vtocl3gf.4	|- (x = A -> (ph <-> ps))
vtocl3gf.5	|- (y = B -> (ps <-> ch))
vtocl3gf.6	|- (z = C -> (ch <-> th))
vtocl3gf.7	|- ph

Assertion

Ref	Expression
vtocl3gf	|- ((A e. V /\ B e. W /\ C e. X) -> th)

Proof of Theorem vtocl3gf

Step	Hyp	Ref	Expression
1		elex 2867	. . 3 |- (A e. V -> A e. _V)
2		vtocl3gf.d	. . . 4 |- F/_yB
3		vtocl3gf.e	. . . 4 |- F/_zB
4		vtocl3gf.f	. . . 4 |- F/_zC
5		vtocl3gf.b	. . . . . 6 |- F/_yA
6	5	nfel1 2499	. . . . 5 |- F/y A e. _V
7		vtocl3gf.2	. . . . 5 |- F/ych
8	6, 7	nfim 1813	. . . 4 |- F/y(A e. _V -> ch)
9		vtocl3gf.c	. . . . . 6 |- F/_zA
10	9	nfel1 2499	. . . . 5 |- F/z A e. _V
11		vtocl3gf.3	. . . . 5 |- F/zth
12	10, 11	nfim 1813	. . . 4 |- F/z(A e. _V -> th)
13		vtocl3gf.5	. . . . 5 |- (y = B -> (ps <-> ch))
14	13	imbi2d 307	. . . 4 |- (y = B -> ((A e. _V -> ps) <-> (A e. _V -> ch)))
15		vtocl3gf.6	. . . . 5 |- (z = C -> (ch <-> th))
16	15	imbi2d 307	. . . 4 |- (z = C -> ((A e. _V -> ch) <-> (A e. _V -> th)))
17		vtocl3gf.a	. . . . 5 |- F/_xA
18		vtocl3gf.1	. . . . 5 |- F/xps
19		vtocl3gf.4	. . . . 5 |- (x = A -> (ph <-> ps))
20		vtocl3gf.7	. . . . 5 |- ph
21	17, 18, 19, 20	vtoclgf 2913	. . . 4 |- (A e. _V -> ps)
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 2916	. . 3 |- ((B e. W /\ C e. X) -> (A e. _V -> th))
23	1, 22	mpan9 455	. 2 |- ((A e. V /\ (B e. W /\ C e. X)) -> th)
24	23	3impb 1147	1 |- ((A e. V /\ B e. W /\ C e. X) -> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476  _Vcvv 2859

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by:  vtocl3gaf  2923