Theorem iunin 3548

Description: Intersection of two complements is equal to the complement of a union. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
iunin	⊢ ∼ (A ∪ B) = ( ∼ A ∩ ∼ B)

Proof of Theorem iunin

Step	Hyp	Ref	Expression
1		dfin5 3546	. 2 ⊢ ( ∼ A ∩ ∼ B) = ∼ ( ∼ ∼ A ∪ ∼ ∼ B)
2		dblcompl 3228	. . . 4 ⊢ ∼ ∼ A = A
3		dblcompl 3228	. . . 4 ⊢ ∼ ∼ B = B
4	2, 3	uneq12i 3417	. . 3 ⊢ ( ∼ ∼ A ∪ ∼ ∼ B) = (A ∪ B)
5	4	compleqi 3245	. 2 ⊢ ∼ ( ∼ ∼ A ∪ ∼ ∼ B) = ∼ (A ∪ B)
6	1, 5	eqtr2i 2374	1 ⊢ ∼ (A ∪ B) = ( ∼ A ∩ ∼ B)

Syntax hints:   = wceq 1642   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215

This theorem is referenced by:  nnadjoin  4521