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Theorem imnot 354
 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), pm2.21 119 (false antecedent), pm5.5 350 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)
Assertion
Ref Expression
imnot 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Proof of Theorem imnot
StepHypRef Expression
1 mtt 353 . 2 𝜓 → (¬ 𝜑 ↔ (𝜑𝜓)))
21bicomd 212 1 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
 Colors of variables: wff setvar class 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:  sup0riota  8254  ntrneikb  37412
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