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Theorem xpeq12d 4809
 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 4803 . 2 ((A = B C = D) → (A × C) = (B × D))
41, 2, 3syl2anc 642 1 (φ → (A × C) = (B × D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   × cxp 4770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-opab 4623  df-xp 4784 This theorem is referenced by:  opeliunxp  4820  fmpt2x  5730
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