Theorem nbn2 334

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		pm5.501 330	. 2 ⊢ (¬ φ → (¬ ψ ↔ (¬ φ ↔ ¬ ψ)))
2		notbi 286	. 2 ⊢ ((φ ↔ ψ) ↔ (¬ φ ↔ ¬ ψ))
3	1, 2	syl6bbr 254	1 ⊢ (¬ φ → (¬ ψ ↔ (φ ↔ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  bibif  335  pm5.21im  338  pm5.18  345  biass  348