Theorem nbn2 373

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		pm5.501 369	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 ↔ ¬ 𝜓)))
2		notbi 322	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	bitr4di 292	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  bibif  374  pm5.21im  377  pm5.18  384  biass  388  sadadd2lem2  16512  isclo  23253