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Theorem eqrelkrdv 4214
 Description: Equality for two Kuratowski relationships. (Contributed by SF, 13-Jan-2015.)
Hypotheses
Ref Expression
eqrelkriiv.1 A (V ×k V)
eqrelkriiv.2 B (V ×k V)
eqrelkrdv.3 (φ → (⟪x, y A ↔ ⟪x, y B))
Assertion
Ref Expression
eqrelkrdv (φA = B)
Distinct variable groups:   x,A,y   x,B,y   φ,x,y

Proof of Theorem eqrelkrdv
StepHypRef Expression
1 eqrelkrdv.3 . . 3 (φ → (⟪x, y A ↔ ⟪x, y B))
21alrimivv 1632 . 2 (φxy(⟪x, y A ↔ ⟪x, y B))
3 eqrelkriiv.1 . . 3 A (V ×k V)
4 eqrelkriiv.2 . . 3 B (V ×k V)
5 eqrelk 4212 . . 3 ((A (V ×k V) B (V ×k V)) → (A = Bxy(⟪x, y A ↔ ⟪x, y B)))
63, 4, 5mp2an 653 . 2 (A = Bxy(⟪x, y A ↔ ⟪x, y B))
72, 6sylibr 203 1 (φA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ⟪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-ral 2619  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: (None)
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