Theorem vtocl3gaf 2750

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
vtocl3gaf.a	|- F/_ x A
vtocl3gaf.b	|- F/_ y A
vtocl3gaf.c	|- F/_ z A
vtocl3gaf.d	|- F/_ y B
vtocl3gaf.e	|- F/_ z B
vtocl3gaf.f	|- F/_ z C
vtocl3gaf.1	|- F/ x ps
vtocl3gaf.2	|- F/ y ch
vtocl3gaf.3	|- F/ z th
vtocl3gaf.4	|- ( x = A -> ( ph <-> ps ) )
vtocl3gaf.5	|- ( y = B -> ( ps <-> ch ) )
vtocl3gaf.6	|- ( z = C -> ( ch <-> th ) )
vtocl3gaf.7	|- ( ( x e. R /\ y e. S /\ z e. T ) -> ph )

Assertion

Ref	Expression
vtocl3gaf	|- ( ( A e. R /\ B e. S /\ C e. T ) -> th )

Distinct variable groups:   x, y, z, R   x, S, y, z   x, T, y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   ch( x, y, z)   th( x, y, z)   A( x, y, z)   B( x, y, z)   C( x, y, z)

Proof of Theorem vtocl3gaf

Step	Hyp	Ref	Expression
1		vtocl3gaf.a	. . 3 |- F/_ x A
2		vtocl3gaf.b	. . 3 |- F/_ y A
3		vtocl3gaf.c	. . 3 |- F/_ z A
4		vtocl3gaf.d	. . 3 |- F/_ y B
5		vtocl3gaf.e	. . 3 |- F/_ z B
6		vtocl3gaf.f	. . 3 |- F/_ z C
7	1	nfel1 2290	. . . . 5 |- F/ x A e. R
8		nfv 1508	. . . . 5 |- F/ x y e. S
9		nfv 1508	. . . . 5 |- F/ x z e. T
10	7, 8, 9	nf3an 1545	. . . 4 |- F/ x ( A e. R /\ y e. S /\ z e. T )
11		vtocl3gaf.1	. . . 4 |- F/ x ps
12	10, 11	nfim 1551	. . 3 |- F/ x ( ( A e. R /\ y e. S /\ z e. T ) -> ps )
13	2	nfel1 2290	. . . . 5 |- F/ y A e. R
14	4	nfel1 2290	. . . . 5 |- F/ y B e. S
15		nfv 1508	. . . . 5 |- F/ y z e. T
16	13, 14, 15	nf3an 1545	. . . 4 |- F/ y ( A e. R /\ B e. S /\ z e. T )
17		vtocl3gaf.2	. . . 4 |- F/ y ch
18	16, 17	nfim 1551	. . 3 |- F/ y ( ( A e. R /\ B e. S /\ z e. T ) -> ch )
19	3	nfel1 2290	. . . . 5 |- F/ z A e. R
20	5	nfel1 2290	. . . . 5 |- F/ z B e. S
21	6	nfel1 2290	. . . . 5 |- F/ z C e. T
22	19, 20, 21	nf3an 1545	. . . 4 |- F/ z ( A e. R /\ B e. S /\ C e. T )
23		vtocl3gaf.3	. . . 4 |- F/ z th
24	22, 23	nfim 1551	. . 3 |- F/ z ( ( A e. R /\ B e. S /\ C e. T ) -> th )
25		eleq1 2200	. . . . 5 |- ( x = A -> ( x e. R <-> A e. R ) )
26	25	3anbi1d 1294	. . . 4 |- ( x = A -> ( ( x e. R /\ y e. S /\ z e. T ) <-> ( A e. R /\ y e. S /\ z e. T ) ) )
27		vtocl3gaf.4	. . . 4 |- ( x = A -> ( ph <-> ps ) )
28	26, 27	imbi12d 233	. . 3 |- ( x = A -> ( ( ( x e. R /\ y e. S /\ z e. T ) -> ph ) <-> ( ( A e. R /\ y e. S /\ z e. T ) -> ps ) ) )
29		eleq1 2200	. . . . 5 |- ( y = B -> ( y e. S <-> B e. S ) )
30	29	3anbi2d 1295	. . . 4 |- ( y = B -> ( ( A e. R /\ y e. S /\ z e. T ) <-> ( A e. R /\ B e. S /\ z e. T ) ) )
31		vtocl3gaf.5	. . . 4 |- ( y = B -> ( ps <-> ch ) )
32	30, 31	imbi12d 233	. . 3 |- ( y = B -> ( ( ( A e. R /\ y e. S /\ z e. T ) -> ps ) <-> ( ( A e. R /\ B e. S /\ z e. T ) -> ch ) ) )
33		eleq1 2200	. . . . 5 |- ( z = C -> ( z e. T <-> C e. T ) )
34	33	3anbi3d 1296	. . . 4 |- ( z = C -> ( ( A e. R /\ B e. S /\ z e. T ) <-> ( A e. R /\ B e. S /\ C e. T ) ) )
35		vtocl3gaf.6	. . . 4 |- ( z = C -> ( ch <-> th ) )
36	34, 35	imbi12d 233	. . 3 |- ( z = C -> ( ( ( A e. R /\ B e. S /\ z e. T ) -> ch ) <-> ( ( A e. R /\ B e. S /\ C e. T ) -> th ) ) )
37		vtocl3gaf.7	. . 3 |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph )
38	1, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37	vtocl3gf 2744	. 2 |- ( ( A e. R /\ B e. S /\ C e. T ) -> ( ( A e. R /\ B e. S /\ C e. T ) -> th ) )
39	38	pm2.43i 49	1 |- ( ( A e. R /\ B e. S /\ C e. T ) -> th )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 962   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2266

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  vtocl3ga  2751