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Theorem bj-notbi 9380
Description: Equivalence property for negation. TODO: minimize all theorems using notbid 591 and notbii 593. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-notbi ((φψ) → (¬ φ ↔ ¬ ψ))

Proof of Theorem bj-notbi
StepHypRef Expression
1 bi2 121 . . 3 ((φψ) → (ψφ))
21con3d 560 . 2 ((φψ) → (¬ φ → ¬ ψ))
3 bi1 111 . . 3 ((φψ) → (φψ))
43con3d 560 . 2 ((φψ) → (¬ ψ → ¬ φ))
52, 4impbid 120 1 ((φψ) → (¬ φ ↔ ¬ ψ))
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:  bj-notbii  9381  bj-notbid  9382  bj-dcbi  9383
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