Theorem ceqsrex2v 3338

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 681	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	rexbii 3041	. . . . 5 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( x = A /\ ( y = B /\ ph ) ) )
3		r19.42v 3092	. . . . 5 |- ( E. y e. D ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
4	2, 3	bitri 264	. . . 4 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
5	4	rexbii 3041	. . 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 740	. . . . 5 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
8	7	rexbidv 3052	. . . 4 |- ( x = A -> ( E. y e. D ( y = B /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
9	8	ceqsrexv 3336	. . 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 272	. 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 3336	. 2 |- ( B e. D -> ( E. y e. D ( y = B /\ ps ) <-> ch ) )
13	10, 12	sylan9bb 736	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rex 2918  df-v 3202

This theorem is referenced by:  opiota  7229  brdom7disj  9353  brdom6disj  9354  lsmspsn  19084