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Theorem rexxp 4950
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
StepHypRef Expression
1 iunxpconst 4867 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21rexeqi 2845 . 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 )
)
43rexiunxp 4948 . 2  |-  ( E. x  e.  U_  y  e.  A  ( {
y }  X.  B
) ph  <->  E. y  e.  A  E. z  e.  B  ps )
52, 4bitr3i 243 1  |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   E.wrex 2643   {csn 3750   <.cop 3753   U_ciun 4028    X. cxp 4809
This theorem is referenced by:  fnrnov  6151  foov  6152  ovelimab  6156  exopxfr  6342  xpf1o  7198  xpwdomg  7479  hsmexlem2  8233  cnref1o  10532  vdwmc  13266  arwhoma  14120  txbas  17513  txkgen  17598  xrofsup  23955  elunirnmbfm  24390  rmxypairf1o  26658  unxpwdom3  26918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-iun 4030  df-opab 4201  df-xp 4817  df-rel 4818
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