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

Step	Hyp	Ref	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
5	2, 3, 4	3eqtr3g 2408	. 2 ⊢ ( ∼ A = ∼ B → A = B)
6	1, 5	impbii 180	1 ⊢ (A = B ↔ ∼ A = ∼ B)

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