Theorem rexxp 4498

Description: Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)

Hypothesis

Ref	Expression
ralxp.1	|- ( x = <. y , z >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexxp	|- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Distinct variable groups:   x, y, z, A   x, B, z   ph, y, z   ps, x   y, B

Allowed substitution hints:   ph( x)   ps( y, z)

Proof of Theorem rexxp

Step	Hyp	Ref	Expression
1		iunxpconst 4418	. . 3 |- U_ y e. A ( { y } X. B ) = ( A X. B )
2	1	rexeqi 2554	. 2 |- ( E. x e. U_ y e. A ( { y } X. B ) ph <-> E. x e. ( A X. B ) ph )
3		ralxp.1	. . 3 |- ( x = <. y , z >. -> ( ph <-> ps ) )
4	3	rexiunxp 4496	. 2 |- ( E. x e. U_ y e. A ( { y } X. B ) ph <-> E. y e. A E. z e. B ps )
5	2, 4	bitr3i 184	1 |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wrex 2349   {csn 3398   <.cop 3401   U_ciun 3678   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  rexxpf  4501  fnrnov  5666  foov  5667  ovelimab  5671  cnref1o  8733