MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfxp Structured version   Unicode version

Theorem nfxp 4904
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 4884 . 2  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
2 nfxp.1 . . . . 5  |-  F/_ x A
32nfcri 2566 . . . 4  |-  F/ x  y  e.  A
4 nfxp.2 . . . . 5  |-  F/_ x B
54nfcri 2566 . . . 4  |-  F/ x  z  e.  B
63, 5nfan 1846 . . 3  |-  F/ x
( y  e.  A  /\  z  e.  B
)
76nfopab 4273 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
81, 7nfcxfr 2569 1  |-  F/_ x
( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725   F/_wnfc 2559   {copab 4265    X. cxp 4876
This theorem is referenced by:  opeliunxp  4929  nfres  5148  mpt2mptsx  6414  dmmpt2ssx  6416  fmpt2x  6417  ovmptss  6428  axcc2  8317  fsum2dlem  12554  fsumcom2  12558  gsumcom2  15549  prdsdsf  18397  prdsxmet  18399  fprod2dlem  25304  fprodcom2  25308  stoweidlem21  27746  stoweidlem47  27772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-opab 4267  df-xp 4884
  Copyright terms: Public domain W3C validator