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Theorem xpeq12i 5135
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 5132 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3mp2an 708 1 (𝐴 × 𝐶) = (𝐵 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482   × cxp 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-opab 4711  df-xp 5118
This theorem is referenced by:  imainrect  5573  cnvssrndm  5655  idssxp  6007  fpar  7278  canthwelem  9469  trclublem  13728  pjpm  20046  txbasval  21403  hausdiag  21442  ussval  22057  ex-xp  27277  hh0oi  28746  fcnvgreu  29457  sitgclg  30389  sitmcl  30398  ismgmOLD  33629  isdrngo1  33735  rtrclex  37750  rtrclexi  37754  trrelsuperrel2dg  37789
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