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Theorem compleqb 3543
 Description: Two classes are equal iff their complements are equal. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
compleqb (A = B ↔ ∼ A = ∼ B)

Proof of Theorem compleqb
StepHypRef Expression
1 compleq 3243 . 2 (A = B → ∼ A = ∼ B)
2 compleq 3243 . . 3 ( ∼ A = ∼ B → ∼ ∼ A = ∼ ∼ B)
3 dblcompl 3227 . . 3 ∼ ∼ A = A
4 dblcompl 3227 . . 3 ∼ ∼ B = B
52, 3, 43eqtr3g 2408 . 2 ( ∼ A = ∼ BA = B)
61, 5impbii 180 1 (A = B ↔ ∼ A = ∼ B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∼ ccompl 3205 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 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-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  nulnnn  4556
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