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Theorem xpeq12d 4613
Description: Equality deduction for Cartesian 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 4607 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3syl2anc 409 1 (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335   × cxp 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-opab 4028  df-xp 4594
This theorem is referenced by:  sqxpeqd  4614  opeliunxp  4643  mpomptsx  6147  dmmpossx  6149  fmpox  6150  disjxp1  6185  erssxp  6505  cc2lem  7188  cc2  7189  fsum2dlemstep  11342  fisumcom2  11346  fprod2dlemstep  11530  fprodcom2fi  11534  txbas  12728
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