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Theorem xpkeq12 4200
 Description: Equality theorem for Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
xpkeq12 ((A = B C = D) → (A ×k C) = (B ×k D))

Proof of Theorem xpkeq12
StepHypRef Expression
1 xpkeq1 4198 . 2 (A = B → (A ×k C) = (B ×k C))
2 xpkeq2 4199 . 2 (C = D → (B ×k C) = (B ×k D))
31, 2sylan9eq 2405 1 ((A = B C = D) → (A ×k C) = (B ×k D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ×k cxpk 4174 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  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-xpk 4185 This theorem is referenced by:  xpkeq12i  4203  xpkeq12d  4206
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