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Theorem nbn2 612
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
nbn2 φ → (¬ ψ ↔ (φψ)))

Proof of Theorem nbn2
StepHypRef Expression
1 pm5.21im 611 . 2 φ → (¬ ψ → (φψ)))
2 bi2 121 . . 3 ((φψ) → (ψφ))
3 mtt 609 . . 3 φ → (¬ ψ ↔ (ψφ)))
42, 3syl5ibr 145 . 2 φ → ((φψ) → ¬ ψ))
51, 4impbid 120 1 φ → (¬ ψ ↔ (φψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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:  bibif  613  pm5.18dc  776  biassdc  1283
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