Theorem nbn 373

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-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 372	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓)
4	3	bicomi 225	1 ⊢ (¬ 𝜓 ↔ (𝜓 ↔ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 207

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 208

This theorem is referenced by:  nbn3  374  nbfal  1562  eq0f  4275  eq0ALT  4279  disj  4378  axnulALT  5226  dm0rn0  5866  dm0rn0OLD  5867  reldm0  5870  isarchi  33263  axnulALT2  35264  axsepg2  35321  axsepg4  35324