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Theorem xpeq12d 4398
Description: Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
xpeq1d.1 (𝜑𝐴 = 𝐵)
xpeq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
xpeq12d (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Proof of Theorem xpeq12d
StepHypRef Expression
1 xpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 xpeq12d.2 . 2 (𝜑𝐶 = 𝐷)
3 xpeq12 4392 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3syl2anc 397 1 (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  opeliunxp  4423  mpt2mptsx  5851  dmmpt2ssx  5853  fmpt2x  5854  erssxp  6160
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