Theorem nbn2 359

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Assertion

Ref	Expression
nbn2	⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))

Proof of Theorem nbn2

Step	Hyp	Ref	Expression
1		pm5.501 355	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 ↔ ¬ 𝜓)))
2		notbi 308	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	syl6bbr 277	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195

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 196

This theorem is referenced by:  bibif  360  pm5.21im  363  pm5.18  370  biass  373  sadadd2lem2  15010  isclo  20701