Theorem nbn2 686

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem nbn2

Step	Hyp	Ref	Expression
1		pm5.21im 685	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓)))
2		bi2 129	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
3		mtt 674	. . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))
4	2, 3	syl5ibr 155	. 2 ⊢ (¬ 𝜑 → ((𝜑 ↔ 𝜓) → ¬ 𝜓))
5	1, 4	impbid 128	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bibif  687  pm5.18dc  868  biassdc  1373