Theorem ceqsrex2v 2812

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

Hypotheses

Ref	Expression
ceqsrex2v.1	|- ( x = A -> ( ph <-> ps ) )
ceqsrex2v.2	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
ceqsrex2v	|- ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )

Distinct variable groups:   x, y, A   x, B, y   x, C   x, D, y   ps, x   ch, y

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

Proof of Theorem ceqsrex2v

Step	Hyp	Ref	Expression
1		anass 398	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	rexbii 2440	. . . . 5 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( x = A /\ ( y = B /\ ph ) ) )
3		r19.42v 2586	. . . . 5 |- ( E. y e. D ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
4	2, 3	bitri 183	. . . 4 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
5	4	rexbii 2440	. . 3 |- ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. x e. C ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
6		ceqsrex2v.1	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
7	6	anbi2d 459	. . . . 5 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
8	7	rexbidv 2436	. . . 4 |- ( x = A -> ( E. y e. D ( y = B /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
9	8	ceqsrexv 2810	. . 3 |- ( A e. C -> ( E. x e. C ( x = A /\ E. y e. D ( y = B /\ ph ) ) <-> E. y e. D ( y = B /\ ps ) ) )
10	5, 9	syl5bb 191	. 2 |- ( A e. C -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
11		ceqsrex2v.2	. . 3 |- ( y = B -> ( ps <-> ch ) )
12	11	ceqsrexv 2810	. 2 |- ( B e. D -> ( E. y e. D ( y = B /\ ps ) <-> ch ) )
13	10, 12	sylan9bb 457	1 |- ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2415

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-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683

This theorem is referenced by: (None)