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Theorem ralxp 4768
Description: Universal quantification restricted to a cross 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 4685 . . 3  |-  U_ y  e.  A  ( {
y }  X.  B
)  =  ( A  X.  B )
21raleqi 2676 . 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 4766 . 2  |-  ( A. x  e.  U_  y  e.  A  ( { y }  X.  B )
ph 
<-> 
A. y  e.  A  A. z  e.  B  ps )
52, 4bitr3i 186 1  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   A.wral 2455   {csn 3592   <.cop 3595   U_ciun 3886    X. cxp 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-iun 3888  df-opab 4064  df-xp 4631  df-rel 4632
This theorem is referenced by:  ralxpf  4771  issref  5009  ffnov  5975  eqfnov  5977  funimassov  6020  f1stres  6156  f2ndres  6157  ecopover  6629  ecopoverg  6632  xpf1o  6840  srgfcl  13078  txbas  13620  cnmpt21  13653  txmetcnp  13880  txmetcn  13881  qtopbasss  13883
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