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Theorem nbn2 360
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
Assertion
Ref Expression
nbn2 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))

Proof of Theorem nbn2
StepHypRef Expression
1 pm5.501 356 . 2 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 ↔ ¬ 𝜓)))
2 notbi 309 . 2 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
31, 2syl6bbr 278 1 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))
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:  bibif  361  pm5.21im  364  pm5.18  371  biass  374  sadadd2lem2  15153  isclo  20872
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