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Theorem nfxp 4536
Description: Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfxp.1  |-  F/_ x A
nfxp.2  |-  F/_ x B
Assertion
Ref Expression
nfxp  |-  F/_ x
( A  X.  B
)

Proof of Theorem nfxp
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4515 . 2  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
2 nfxp.1 . . . . 5  |-  F/_ x A
32nfcri 2252 . . . 4  |-  F/ x  y  e.  A
4 nfxp.2 . . . . 5  |-  F/_ x B
54nfcri 2252 . . . 4  |-  F/ x  z  e.  B
63, 5nfan 1529 . . 3  |-  F/ x
( y  e.  A  /\  z  e.  B
)
76nfopab 3966 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
81, 7nfcxfr 2255 1  |-  F/_ x
( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1465   F/_wnfc 2245   {copab 3958    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-opab 3960  df-xp 4515
This theorem is referenced by:  opeliunxp  4564  nfres  4791  mpomptsx  6063  dmmpossx  6065  fmpox  6066  disjxp1  6101  nfdju  6895  fsum2dlemstep  11171  fisumcom2  11175
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