Theorem vtocl2gaf 3174

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
vtocl2gaf.a	|- F/_ x A
vtocl2gaf.b	|- F/_ y A
vtocl2gaf.c	|- F/_ y B
vtocl2gaf.1	|- F/ x ps
vtocl2gaf.2	|- F/ y ch
vtocl2gaf.3	|- ( x = A -> ( ph <-> ps ) )
vtocl2gaf.4	|- ( y = B -> ( ps <-> ch ) )
vtocl2gaf.5	|- ( ( x e. C /\ y e. D ) -> ph )

Assertion

Ref	Expression
vtocl2gaf	|- ( ( A e. C /\ B e. D ) -> ch )

Distinct variable groups:   x, y, C   x, D, y

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

Proof of Theorem vtocl2gaf

Step	Hyp	Ref	Expression
1		vtocl2gaf.a	. . 3 |- F/_ x A
2		vtocl2gaf.b	. . 3 |- F/_ y A
3		vtocl2gaf.c	. . 3 |- F/_ y B
4	1	nfel1 2635	. . . . 5 |- F/ x A e. C
5		nfv 1708	. . . . 5 |- F/ x y e. D
6	4, 5	nfan 1929	. . . 4 |- F/ x ( A e. C /\ y e. D )
7		vtocl2gaf.1	. . . 4 |- F/ x ps
8	6, 7	nfim 1921	. . 3 |- F/ x ( ( A e. C /\ y e. D ) -> ps )
9	2	nfel1 2635	. . . . 5 |- F/ y A e. C
10	3	nfel1 2635	. . . . 5 |- F/ y B e. D
11	9, 10	nfan 1929	. . . 4 |- F/ y ( A e. C /\ B e. D )
12		vtocl2gaf.2	. . . 4 |- F/ y ch
13	11, 12	nfim 1921	. . 3 |- F/ y ( ( A e. C /\ B e. D ) -> ch )
14		eleq1 2529	. . . . 5 |- ( x = A -> ( x e. C <-> A e. C ) )
15	14	anbi1d 704	. . . 4 |- ( x = A -> ( ( x e. C /\ y e. D ) <-> ( A e. C /\ y e. D ) ) )
16		vtocl2gaf.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
17	15, 16	imbi12d 320	. . 3 |- ( x = A -> ( ( ( x e. C /\ y e. D ) -> ph ) <-> ( ( A e. C /\ y e. D ) -> ps ) ) )
18		eleq1 2529	. . . . 5 |- ( y = B -> ( y e. D <-> B e. D ) )
19	18	anbi2d 703	. . . 4 |- ( y = B -> ( ( A e. C /\ y e. D ) <-> ( A e. C /\ B e. D ) ) )
20		vtocl2gaf.4	. . . 4 |- ( y = B -> ( ps <-> ch ) )
21	19, 20	imbi12d 320	. . 3 |- ( y = B -> ( ( ( A e. C /\ y e. D ) -> ps ) <-> ( ( A e. C /\ B e. D ) -> ch ) ) )
22		vtocl2gaf.5	. . 3 |- ( ( x e. C /\ y e. D ) -> ph )
23	1, 2, 3, 8, 13, 17, 21, 22	vtocl2gf 3169	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( A e. C /\ B e. D ) -> ch ) )
24	23	pm2.43i 47	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   F/wnf 1617   e. wcel 1819   F/_wnfc 2605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111

This theorem is referenced by:  vtocl2ga  3175  ovmpt2s  6425  ov2gf  6426  ov3  6438  pwfseqlem2  9054  cnmptcom  20305