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Theorem nbn 614
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 613 . . 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  615  nbfal  1253  n0rf  3227  eq0  3233  disj  3262  dm0rn0  4495  reldm0  4496
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