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Theorem nbn 625
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 624 . . 3 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
31, 2ax-mp 7 . 2 ((𝜓𝜑) ↔ ¬ 𝜓)
43bicomi 127 1 𝜓 ↔ (𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  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:  nbn3  626  nbfal  1270  n0rf  3261  eq0  3267  disj  3296  dm0rn0  4580  reldm0  4581
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