Theorem nbn 699

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 698	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓)
4	3	bicomi 132	1 ⊢ (¬ 𝜓 ↔ (𝜓 ↔ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nbn3  700  nbfal  1364  n0rf  3436  eq0  3442  disj  3472  dm0rn0  4845  reldm0  4846