Theorem imnot 365

Description: If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication (𝜑 → 𝜓), the other ones being ax-1 6 (true consequent), pm2.21 123 (false antecedent), pm5.5 361 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)

Assertion

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

Proof of Theorem imnot

Step	Hyp	Ref	Expression
1		mtt 364	. 2 ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜑 → 𝜓)))
2	1	bicomd 223	1 ⊢ (¬ 𝜓 → ((𝜑 → 𝜓) ↔ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

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 207

This theorem is referenced by:  sup0riota  9483  ntrneikb  44085