Theorem vtocl3gf 2672

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/_ x A
vtocl3gf.b	|- F/_ y A
vtocl3gf.c	|- F/_ z A
vtocl3gf.d	|- F/_ y B
vtocl3gf.e	|- F/_ z B
vtocl3gf.f	|- F/_ z C
vtocl3gf.1	|- F/ x ps
vtocl3gf.2	|- F/ y ch
vtocl3gf.3	|- F/ z th
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 2621	. . 3 |- ( A e. V -> A e. _V )
2		vtocl3gf.d	. . . 4 |- F/_ y B
3		vtocl3gf.e	. . . 4 |- F/_ z B
4		vtocl3gf.f	. . . 4 |- F/_ z C
5		vtocl3gf.b	. . . . . 6 |- F/_ y A
6	5	nfel1 2233	. . . . 5 |- F/ y A e. _V
7		vtocl3gf.2	. . . . 5 |- F/ y ch
8	6, 7	nfim 1505	. . . 4 |- F/ y ( A e. _V -> ch )
9		vtocl3gf.c	. . . . . 6 |- F/_ z A
10	9	nfel1 2233	. . . . 5 |- F/ z A e. _V
11		vtocl3gf.3	. . . . 5 |- F/ z th
12	10, 11	nfim 1505	. . . 4 |- F/ z ( A e. _V -> th )
13		vtocl3gf.5	. . . . 5 |- ( y = B -> ( ps <-> ch ) )
14	13	imbi2d 228	. . . 4 |- ( y = B -> ( ( A e. _V -> ps ) <-> ( A e. _V -> ch ) ) )
15		vtocl3gf.6	. . . . 5 |- ( z = C -> ( ch <-> th ) )
16	15	imbi2d 228	. . . 4 |- ( z = C -> ( ( A e. _V -> ch ) <-> ( A e. _V -> th ) ) )
17		vtocl3gf.a	. . . . 5 |- F/_ x A
18		vtocl3gf.1	. . . . 5 |- F/ x ps
19		vtocl3gf.4	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
20		vtocl3gf.7	. . . . 5 |- ph
21	17, 18, 19, 20	vtoclgf 2668	. . . 4 |- ( A e. _V -> ps )
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 2671	. . 3 |- ( ( B e. W /\ C e. X ) -> ( A e. _V -> th ) )
23	1, 22	mpan9 275	. 2 |- ( ( A e. V /\ ( B e. W /\ C e. X ) ) -> th )
24	23	3impb 1135	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> th )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   F/wnf 1390   e. wcel 1434   F/_wnfc 2210   _Vcvv 2612

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614

This theorem is referenced by:  vtocl3gaf  2678