Theorem rexxp 4755

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 4671	. . 3 |- U_ y e. A ( { y } X. B ) = ( A X. B )
2	1	rexeqi 2670	. 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 4753	. 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 185	1 |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wrex 2449   {csn 3583   <.cop 3586   U_ciun 3873   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  rexxpf  4758  fnrnov  5998  foov  5999  ovelimab  6003  xpf1o  6822  cnref1o  9609  txbas  13052