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Theorem ralxp 5135
Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
ralxp  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. 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 ralxp
StepHypRef Expression
1 iunxpconst 5048 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21raleqi 3055 . 2  |-  ( A. x  e.  U_  y  e.  A  ( { y }  X.  B )
ph 
<-> 
A. x  e.  ( A  X.  B )
ph )
3 ralxp.1 . . 3  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
43raliunxp 5133 . 2  |-  ( A. x  e.  U_  y  e.  A  ( { y }  X.  B )
ph 
<-> 
A. y  e.  A  A. z  e.  B  ps )
52, 4bitr3i 251 1  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   A.wral 2807   {csn 4020   <.cop 4026   U_ciun 4318    X. cxp 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-iun 4320  df-opab 4499  df-xp 4998  df-rel 4999
This theorem is referenced by:  ralxpf  5140  issref  5371  ffnov  6381  eqfnov  6383  funimassov  6427  f1stres  6796  f2ndres  6797  ecopover  7405  xpf1o  7669  xpwdomg  8000  rankxplim  8286  imasaddfnlem  14772  imasvscafn  14781  comfeq  14951  isssc  15039  isfuncd  15081  cofucl  15104  funcres2b  15113  evlfcl  15338  uncfcurf  15355  yonedalem3  15396  yonedainv  15397  efgval2  16531  txbas  19796  hausdiag  19874  tx1stc  19879  txkgen  19881  xkococn  19889  cnmpt21  19900  xkoinjcn  19916  tmdcn2  20316  clssubg  20335  divstgplem  20347  txmetcnp  20778  txmetcn  20779  qtopbaslem  20993  bndth  21186  cxpcn3  22843  dvdsmulf1o  23191  fsumdvdsmul  23192  xrofsup  27236  txpcon  28303  cvmlift2lem1  28373  cvmlift2lem12  28385  f1opr  29805  ismtyhmeolem  29890  ffnaov  31706  dih1dimatlem  36001
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