Theorem rexxp 4822

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 4735	. . 3 |- U_ y e. A ( { y } X. B ) = ( A X. B )
2	1	rexeqi 2707	. 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 4820	. 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 186	1 |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wrex 2485   {csn 3633   <.cop 3636   U_ciun 3927   X. cxp 4673

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-iun 3929  df-opab 4106  df-xp 4681  df-rel 4682

This theorem is referenced by:  rexxpf  4825  fnrnov  6092  foov  6093  ovelimab  6097  xpf1o  6941  cnref1o  9772  txbas  14730