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Theorem compleq 3714
 Description: Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
compleq (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))

Proof of Theorem compleq
StepHypRef Expression
1 complss 3713 . . 3 (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
2 complss 3713 . . 3 (𝐵𝐴 ↔ (V ∖ 𝐴) ⊆ (V ∖ 𝐵))
31, 2anbi12ci 730 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
4 eqss 3583 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3583 . 2 ((V ∖ 𝐴) = (V ∖ 𝐵) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
63, 4, 53bitr4i 291 1 (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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