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Theorem nbn 374
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
Hypothesis
Ref Expression
nbn.1 ¬ 𝜑
Assertion
Ref Expression
nbn 𝜓 ↔ (𝜓𝜑))

Proof of Theorem nbn
StepHypRef Expression
1 nbn.1 . . 3 ¬ 𝜑
2 bibif 373 . . 3 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
31, 2ax-mp 5 . 2 ((𝜓𝜑) ↔ ¬ 𝜓)
43bicomi 225 1 𝜓 ↔ (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207
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 208
This theorem is referenced by:  nbn3  375  nbfal  1543  eq0f  4302  disj  4395  axnulALT  5199  dm0rn0  5788  reldm0  5791  isarchi  30738
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