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Theorem ralxp 5133
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 5045 . . 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 5131 . 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 1398   A.wral 2804   {csn 4016   <.cop 4022   U_ciun 4315    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iun 4317  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by:  ralxpf  5138  issref  5368  ffnov  6379  eqfnov  6381  funimassov  6425  f1stres  6795  f2ndres  6796  ecopover  7407  xpf1o  7672  xpwdomg  8003  rankxplim  8288  imasaddfnlem  15020  imasvscafn  15029  comfeq  15197  isssc  15311  isfuncd  15356  cofucl  15379  funcres2b  15388  evlfcl  15693  uncfcurf  15710  yonedalem3  15751  yonedainv  15752  efgval2  16944  txbas  20237  hausdiag  20315  tx1stc  20320  txkgen  20322  xkococn  20330  cnmpt21  20341  xkoinjcn  20357  tmdcn2  20757  clssubg  20776  qustgplem  20788  txmetcnp  21219  txmetcn  21220  qtopbaslem  21434  bndth  21627  cxpcn3  23293  dvdsmulf1o  23671  fsumdvdsmul  23672  xrofsup  27819  txpcon  28944  cvmlift2lem1  29014  cvmlift2lem12  29026  mclsax  29196  f1opr  30458  ismtyhmeolem  30543  ffnaov  32526  ovn0ssdmfun  32846  plusfreseq  32851  dih1dimatlem  37472
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