Theorem vtocl3gf 2793

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 2741	. . 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 2323	. . . . 5 |- F/ y A e. _V
7		vtocl3gf.2	. . . . 5 |- F/ y ch
8	6, 7	nfim 1565	. . . 4 |- F/ y ( A e. _V -> ch )
9		vtocl3gf.c	. . . . . 6 |- F/_ z A
10	9	nfel1 2323	. . . . 5 |- F/ z A e. _V
11		vtocl3gf.3	. . . . 5 |- F/ z th
12	10, 11	nfim 1565	. . . 4 |- F/ z ( A e. _V -> th )
13		vtocl3gf.5	. . . . 5 |- ( y = B -> ( ps <-> ch ) )
14	13	imbi2d 229	. . . 4 |- ( y = B -> ( ( A e. _V -> ps ) <-> ( A e. _V -> ch ) ) )
15		vtocl3gf.6	. . . . 5 |- ( z = C -> ( ch <-> th ) )
16	15	imbi2d 229	. . . 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 2788	. . . 4 |- ( A e. _V -> ps )
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 2792	. . 3 |- ( ( B e. W /\ C e. X ) -> ( A e. _V -> th ) )
23	1, 22	mpan9 279	. 2 |- ( ( A e. V /\ ( B e. W /\ C e. X ) ) -> th )
24	23	3impb 1194	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> th )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   F/wnf 1453   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  vtocl3gaf  2799