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Theorem rexxp 4994
Description: Existential quantification restricted to a Cartesian 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 4907 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21rexeqi 2934 . 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 4992 . 2  |-  ( E. x  e.  U_  y  e.  A  ( {
y }  X.  B
) ph  <->  E. y  e.  A  E. z  e.  B  ps )
52, 4bitr3i 251 1  |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369   E.wrex 2728   {csn 3889   <.cop 3895   U_ciun 4183    X. cxp 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-iun 4185  df-opab 4363  df-xp 4858  df-rel 4859
This theorem is referenced by:  exopxfr  4995  fnrnov  6248  foov  6249  ovelimab  6253  xpf1o  7485  xpwdomg  7812  hsmexlem2  8608  cnref1o  10998  vdwmc  14051  arwhoma  14925  txbas  19152  txkgen  19237  xrofsup  26067  elunirnmbfm  26680  rmxypairf1o  29264  unxpwdom3  29460  el2xptp  30138
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