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Theorem xpkeq2 4199
 Description: Equality theorem for Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
xpkeq2 (A = B → (C ×k A) = (C ×k B))

Proof of Theorem xpkeq2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2808 . . . 4 (A = B → (z A x = ⟪y, z⟫ ↔ z B x = ⟪y, z⟫))
21rexbidv 2635 . . 3 (A = B → (y C z A x = ⟪y, z⟫ ↔ y C z B x = ⟪y, z⟫))
3 elxpk2 4197 . . 3 (x (C ×k A) ↔ y C z A x = ⟪y, z⟫)
4 elxpk2 4197 . . 3 (x (C ×k B) ↔ y C z B x = ⟪y, z⟫)
52, 3, 43bitr4g 279 . 2 (A = B → (x (C ×k A) ↔ x (C ×k B)))
65eqrdv 2351 1 (A = B → (C ×k A) = (C ×k B))
 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-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:  xpkeq12  4200  xpkeq2i  4202  xpkeq2d  4205  xpkvexg  4285
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