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Theorem xpeq12d 4663
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 4657 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
41, 2, 3syl2anc 411 1 (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363   × cxp 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-opab 4077  df-xp 4644
This theorem is referenced by:  sqxpeqd  4664  opeliunxp  4693  mpomptsx  6211  dmmpossx  6213  fmpox  6214  disjxp1  6250  erssxp  6571  cc2lem  7278  cc2  7279  fsum2dlemstep  11455  fisumcom2  11459  fprod2dlemstep  11643  fprodcom2fi  11647  txbas  14029
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