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Theorem nfxp 4454
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 4434 . 2  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
2 nfxp.1 . . . . 5  |-  F/_ x A
32nfcri 2222 . . . 4  |-  F/ x  y  e.  A
4 nfxp.2 . . . . 5  |-  F/_ x B
54nfcri 2222 . . . 4  |-  F/ x  z  e.  B
63, 5nfan 1502 . . 3  |-  F/ x
( y  e.  A  /\  z  e.  B
)
76nfopab 3898 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
81, 7nfcxfr 2225 1  |-  F/_ x
( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1438   F/_wnfc 2215   {copab 3890    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-opab 3892  df-xp 4434
This theorem is referenced by:  opeliunxp  4481  nfres  4703  mpt2mptsx  5949  dmmpt2ssx  5951  fmpt2x  5952  disjxp1  5983  nfdju  6714  fsum2dlemstep  10791  fisumcom2  10795
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