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Theorem iunin 3547
 Description: Intersection of two complements is equal to the complement of a union. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
iunin ∼ (AB) = ( ∼ A ∩ ∼ B)

Proof of Theorem iunin
StepHypRef Expression
1 dfin5 3545 . 2 ( ∼ A ∩ ∼ B) = ∼ ( ∼ ∼ A ∪ ∼ ∼ B)
2 dblcompl 3227 . . . 4 ∼ ∼ A = A
3 dblcompl 3227 . . . 4 ∼ ∼ B = B
42, 3uneq12i 3416 . . 3 ( ∼ ∼ A ∪ ∼ ∼ B) = (AB)
54compleqi 3244 . 2 ∼ ( ∼ ∼ A ∪ ∼ ∼ B) = ∼ (AB)
61, 5eqtr2i 2374 1 ∼ (AB) = ( ∼ A ∩ ∼ B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208 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-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  df-in 3213  df-un 3214 This theorem is referenced by:  nnadjoin  4520
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