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Theorem nbn2 646
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 ↔ (𝜑𝜓)))

Proof of Theorem nbn2
StepHypRef Expression
1 pm5.21im 645 . 2 𝜑 → (¬ 𝜓 → (𝜑𝜓)))
2 bi2 128 . . 3 ((𝜑𝜓) → (𝜓𝜑))
3 mtt 643 . . 3 𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
42, 3syl5ibr 154 . 2 𝜑 → ((𝜑𝜓) → ¬ 𝜓))
51, 4impbid 127 1 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bibif  647  pm5.18dc  811  biassdc  1327
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