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Theorem bibif 624
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 623 . 2 𝜓 → (¬ 𝜑 ↔ (𝜓𝜑)))
2 bicom 132 . 2 ((𝜓𝜑) ↔ (𝜑𝜓))
31, 2syl6rbb 190 1 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  nbn  625
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