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Theorem nbn 615
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-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 614 . . 3 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
31, 2ax-mp 7 . 2 ((𝜓𝜑) ↔ ¬ 𝜓)
43bicomi 123 1 𝜓 ↔ (𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  nbn3  616  nbfal  1254  n0rf  3233  eq0  3239  disj  3268  dm0rn0  4552  reldm0  4553
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