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Theorem xpeq12i 5713
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 5710 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3mp2an 692 1 (𝐴 × 𝐶) = (𝐵 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-opab 5206  df-xp 5691
This theorem is referenced by:  imainrect  6201  cnvrescnv  6215  cnvssrndm  6291  fpar  8141  ttrclexg  9763  canthwelem  10690  trclublem  15034  pjpm  21728  txbasval  23614  hausdiag  23653  ussval  24268  ex-xp  30455  hh0oi  31922  fcnvgreu  32683  sitgclg  34344  sitmcl  34353  ismgmOLD  37857  isdrngo1  37963  trrelsuperrel2dg  43684
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