Theorem rexxp 4840

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 4753	. . 3 |- U_ y e. A ( { y } X. B ) = ( A X. B )
2	1	rexeqi 2710	. 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 4838	. 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 2487   {csn 3643   <.cop 3646   U_ciun 3941   X. cxp 4691

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-iun 3943  df-opab 4122  df-xp 4699  df-rel 4700

This theorem is referenced by:  rexxpf  4843  fnrnov  6115  foov  6116  ovelimab  6120  xpf1o  6966  cnref1o  9807  txbas  14845