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Theorem xpeq12i 5021
Description: Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.)
Hypotheses
Ref Expression
xpeq12i.1  |-  A  =  B
xpeq12i.2  |-  C  =  D
Assertion
Ref Expression
xpeq12i  |-  ( A  X.  C )  =  ( B  X.  D
)

Proof of Theorem xpeq12i
StepHypRef Expression
1 xpeq12i.1 . 2  |-  A  =  B
2 xpeq12i.2 . 2  |-  C  =  D
3 xpeq12 5018 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
41, 2, 3mp2an 672 1  |-  ( A  X.  C )  =  ( B  X.  D
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-opab 4506  df-xp 5005
This theorem is referenced by:  xpssres  5308  imainrect  5448  cnvssrndm  5529  fpar  6888  canthwelem  9029  pjpm  18546  txbasval  19934  hausdiag  19973  ussval  20589  ex-xp  24931  ismgm  25095  ghsubgolem  25145  hh0oi  26595  idssxp  27239  fcnvgreu  27283  sitgclg  28035  isdrngo1  30189  trclubg  37012
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