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Theorem cnvkeq 4215
 Description: Equality theorem for Kuratowski converse. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
cnvkeq (A = BkA = kB)

Proof of Theorem cnvkeq
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . 5 (A = B → (⟪z, y A ↔ ⟪z, y B))
21anbi2d 684 . . . 4 (A = B → ((x = ⟪y, zz, y A) ↔ (x = ⟪y, zz, y B)))
322exbidv 1628 . . 3 (A = B → (yz(x = ⟪y, zz, y A) ↔ yz(x = ⟪y, zz, y B)))
43abbidv 2467 . 2 (A = B → {x yz(x = ⟪y, zz, y A)} = {x yz(x = ⟪y, zz, y B)})
5 df-cnvk 4186 . 2 kA = {x yz(x = ⟪y, zz, y A)}
6 df-cnvk 4186 . 2 kB = {x yz(x = ⟪y, zz, y B)}
74, 5, 63eqtr4g 2410 1 (A = BkA = kB)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟪copk 4057  ◡kccnvk 4175 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-cnvk 4186 This theorem is referenced by:  cnvkeqi  4216  cnvkeqd  4217  cokeq2  4231  cnvkexg  4286
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