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Theorem bibif 360
Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
Assertion
Ref Expression
bibif 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Proof of Theorem bibif
StepHypRef Expression
1 nbn2 359 . 2 𝜓 → (¬ 𝜑 ↔ (𝜓𝜑)))
2 bicom 212 . 2 ((𝜓𝜑) ↔ (𝜑𝜓))
31, 2syl6rbb 277 1 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196
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 197
This theorem is referenced by:  nbn  361  bj-bibibi  32877  or3or  38821
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