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Theorem nfxp 4313
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 xA
nfxp.2 xB
Assertion
Ref Expression
nfxp x(A × B)

Proof of Theorem nfxp
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4293 . 2 (A × B) = {⟨y, z⟩ ∣ (y A z B)}
2 nfxp.1 . . . . 5 xA
32nfcri 2169 . . . 4 x y A
4 nfxp.2 . . . . 5 xB
54nfcri 2169 . . . 4 x z B
63, 5nfan 1454 . . 3 x(y A z B)
76nfopab 3815 . 2 x{⟨y, z⟩ ∣ (y A z B)}
81, 7nfcxfr 2172 1 x(A × B)
Colors of variables: wff set class
Syntax hints:   wa 97   wcel 1390  wnfc 2162  {copab 3807   × cxp 4285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-opab 3809  df-xp 4293
This theorem is referenced by:  opeliunxp  4337  nfres  4556  mpt2mptsx  5762  dmmpt2ssx  5764  fmpt2x  5765
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