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Theorem xpkeq1 4198
 Description: Equality theorem for Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
xpkeq1 (A = B → (A ×k C) = (B ×k C))

Proof of Theorem xpkeq1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2808 . . 3 (A = B → (y A z C x = ⟪y, z⟫ ↔ y B z C x = ⟪y, z⟫))
2 elxpk2 4197 . . 3 (x (A ×k C) ↔ y A z C x = ⟪y, z⟫)
3 elxpk2 4197 . . 3 (x (B ×k C) ↔ y B z C x = ⟪y, z⟫)
41, 2, 33bitr4g 279 . 2 (A = B → (x (A ×k C) ↔ x (B ×k C)))
54eqrdv 2351 1 (A = B → (A ×k C) = (B ×k C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟪copk 4057   ×k cxpk 4174 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  xpkeq12  4200  xpkeq1i  4201  xpkeq1d  4204
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