Theorem ceqsrex2v 2958

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 405	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	rexbii 2557	. . . . 5 |- ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( x = A /\ ( y = B /\ ph ) ) )
3		r19.42v 2708	. . . . 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 2557	. . 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 468	. . . . 5 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
8	7	rexbidv 2551	. . . 4 |- ( x = A -> ( E. y e. D ( y = B /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
9	8	ceqsrexv 2956	. . 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 2956	. 2 |- ( B e. D -> ( E. y e. D ( y = B /\ ps ) <-> ch ) )
13	10, 12	sylan9bb 466	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 1402   e. wcel 2209   E.wrex 2529

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823

This theorem is referenced by: (None)