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Theorem xpeq12i 5710
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 × 𝐶) = (𝐵 × 𝐷)

Proof of Theorem xpeq12i
StepHypRef Expression
1 xpeq12i.1 . 2 𝐴 = 𝐵
2 xpeq12i.2 . 2 𝐶 = 𝐷
3 xpeq12 5707 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3mp2an 690 1 (𝐴 × 𝐶) = (𝐵 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534   × cxp 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-opab 5216  df-xp 5688
This theorem is referenced by:  imainrect  6192  cnvrescnv  6206  cnvssrndm  6282  fpar  8130  ttrclexg  9766  canthwelem  10693  trclublem  15000  pjpm  21706  txbasval  23601  hausdiag  23640  ussval  24255  ex-xp  30369  hh0oi  31836  fcnvgreu  32590  sitgclg  34176  sitmcl  34185  ismgmOLD  37551  isdrngo1  37657  trrelsuperrel2dg  43338
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