Theorem ceqsrex2v 2892

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 401	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	rexbii 2501	. . . . 5 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( x = A /\ ( y = B /\ ph ) ) )
3		r19.42v 2651	. . . . 5 |- ( E. y e. D ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
4	2, 3	bitri 184	. . . 4 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
5	4	rexbii 2501	. . 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 464	. . . . 5 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
8	7	rexbidv 2495	. . . 4 |- ( x = A -> ( E. y e. D ( y = B /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
9	8	ceqsrexv 2890	. . 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	bitrid 192	. 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 2890	. 2 |- ( B e. D -> ( E. y e. D ( y = B /\ ps ) <-> ch ) )
13	10, 12	sylan9bb 462	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 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762

This theorem is referenced by: (None)