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Theorem xpeq12i 5566
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 5563 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3mp2an 691 1 (𝐴 × 𝐶) = (𝐵 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538   × cxp 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-opab 5112  df-xp 5544
This theorem is referenced by:  imainrect  6021  cnvrescnv  6035  cnvssrndm  6105  fpar  7796  canthwelem  10059  trclublem  14346  pjpm  20840  txbasval  22202  hausdiag  22241  ussval  22856  ex-xp  28212  hh0oi  29677  fcnvgreu  30417  sitgclg  31620  sitmcl  31629  ismgmOLD  35193  isdrngo1  35299  trrelsuperrel2dg  40219
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