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Theorem xpeq12i 4856
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 4853 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
41, 2, 3mp2an 678 1  |-  ( A  X.  C )  =  ( B  X.  D
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444    X. cxp 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-opab 4462  df-xp 4840
This theorem is referenced by:  xpssres  5139  imainrect  5278  cnvssrndm  5357  fpar  6900  canthwelem  9075  trclublem  13059  pjpm  19271  txbasval  20621  hausdiag  20660  ussval  21274  ex-xp  25886  ismgmOLD  26048  ghsubgolemOLD  26098  hh0oi  27556  idssxp  28227  fcnvgreu  28275  sitgclg  29175  sitmcl  29184  isdrngo1  32195  rtrclex  36224  rtrclexi  36228  trrelsuperrel2dg  36263
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