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Theorem nfxp 4668
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 4647 . 2  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
2 nfxp.1 . . . . 5  |-  F/_ x A
32nfcri 2326 . . . 4  |-  F/ x  y  e.  A
4 nfxp.2 . . . . 5  |-  F/_ x B
54nfcri 2326 . . . 4  |-  F/ x  z  e.  B
63, 5nfan 1576 . . 3  |-  F/ x
( y  e.  A  /\  z  e.  B
)
76nfopab 4086 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
81, 7nfcxfr 2329 1  |-  F/_ x
( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2160   F/_wnfc 2319   {copab 4078    X. cxp 4639
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-opab 4080  df-xp 4647
This theorem is referenced by:  opeliunxp  4696  nfres  4924  mpomptsx  6217  dmmpossx  6219  fmpox  6220  disjxp1  6256  nfdju  7066  fsum2dlemstep  11469  fisumcom2  11473  fprod2dlemstep  11657  fprodcom2fi  11661
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