MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexxp Structured version   Unicode version

Theorem rexxp 5020
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 4937 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21rexeqi 2911 . 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 5018 . 2  |-  ( E. x  e.  U_  y  e.  A  ( {
y }  X.  B
) ph  <->  E. y  e.  A  E. z  e.  B  ps )
52, 4bitr3i 244 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 178    = wceq 1653   E.wrex 2708   {csn 3816   <.cop 3819   U_ciun 4095    X. cxp 4879
This theorem is referenced by:  fnrnov  6222  foov  6223  ovelimab  6227  exopxfr  6413  xpf1o  7272  xpwdomg  7556  hsmexlem2  8312  cnref1o  10612  vdwmc  13351  arwhoma  14205  txbas  17604  txkgen  17689  xrofsup  24131  elunirnmbfm  24608  rmxypairf1o  26988  unxpwdom3  27247  el2xptp  28075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-iun 4097  df-opab 4270  df-xp 4887  df-rel 4888
  Copyright terms: Public domain W3C validator