Theorem nbn 336

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Hypothesis

Ref	Expression
nbn.1	⊢ ¬ φ

Assertion

Ref	Expression
nbn	⊢ (¬ ψ ↔ (ψ ↔ φ))

Proof of Theorem nbn

Step	Hyp	Ref	Expression
1		nbn.1	. . 3 ⊢ ¬ φ
2		bibif 335	. . 3 ⊢ (¬ φ → ((ψ ↔ φ) ↔ ¬ ψ))
3	1, 2	ax-mp 5	. 2 ⊢ ((ψ ↔ φ) ↔ ¬ ψ)
4	3	bicomi 193	1 ⊢ (¬ ψ ↔ (ψ ↔ φ))

Syntax hints:  ¬ wn 3   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  nbn3  337  nbfal  1325  n0f  3559  dm0rn0  4922  dmeq0  4923