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Theorem xpeq12i 4395
Description: Equality inference for cross 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 4392 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3mp2an 410 1 (𝐴 × 𝐶) = (𝐵 × 𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1259   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-opab 3847  df-xp 4379
This theorem is referenced by:  xpssres  4673  imainrect  4794  cnvssrndm  4870
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