Theorem vtocl2gaf 2753

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 2292	. . . . 5 |- F/ x A e. C
5		nfv 1508	. . . . 5 |- F/ x y e. D
6	4, 5	nfan 1544	. . . 4 |- F/ x ( A e. C /\ y e. D )
7		vtocl2gaf.1	. . . 4 |- F/ x ps
8	6, 7	nfim 1551	. . 3 |- F/ x ( ( A e. C /\ y e. D ) -> ps )
9	2	nfel1 2292	. . . . 5 |- F/ y A e. C
10	3	nfel1 2292	. . . . 5 |- F/ y B e. D
11	9, 10	nfan 1544	. . . 4 |- F/ y ( A e. C /\ B e. D )
12		vtocl2gaf.2	. . . 4 |- F/ y ch
13	11, 12	nfim 1551	. . 3 |- F/ y ( ( A e. C /\ B e. D ) -> ch )
14		eleq1 2202	. . . . 5 |- ( x = A -> ( x e. C <-> A e. C ) )
15	14	anbi1d 460	. . . 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 233	. . 3 |- ( x = A -> ( ( ( x e. C /\ y e. D ) -> ph ) <-> ( ( A e. C /\ y e. D ) -> ps ) ) )
18		eleq1 2202	. . . . 5 |- ( y = B -> ( y e. D <-> B e. D ) )
19	18	anbi2d 459	. . . 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 233	. . 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 2748	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( A e. C /\ B e. D ) -> ch ) )
24	23	pm2.43i 49	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2268

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  vtocl2ga  2754  ovmpos  5894  ov2gf  5895  ovi3  5907  cnmptcom  12467