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Theorem rexxp 4480
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 4400 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21rexeqi 2510 . 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 4478 . 2  |-  ( E. x  e.  U_  y  e.  A  ( {
y }  X.  B
) ph  <->  E. y  e.  A  E. z  e.  B  ps )
52, 4bitr3i 175 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 98    = wceq 1243   E.wrex 2307   {csn 3375   <.cop 3378   U_ciun 3657    X. cxp 4343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-xp 4351  df-rel 4352
This theorem is referenced by:  rexxpf  4483  fnrnov  5646  foov  5647  ovelimab  5651  cnref1o  8582
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