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Theorem xpeq12i 4867
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 4864 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
41, 2, 3mp2an 676 1  |-  ( A  X.  C )  =  ( B  X.  D
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    X. cxp 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-opab 4476  df-xp 4851
This theorem is referenced by:  xpssres  5150  imainrect  5289  cnvssrndm  5368  fpar  6902  canthwelem  9064  trclublem  13027  pjpm  19208  txbasval  20558  hausdiag  20597  ussval  21211  ex-xp  25772  ismgmOLD  25934  ghsubgolemOLD  25984  hh0oi  27432  idssxp  28108  fcnvgreu  28156  sitgclg  29044  isdrngo1  31943  trrelsuperrel2dg  35950
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