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Theorem bj-notbi 10983
Description: Equivalence property for negation. TODO: minimize all theorems using notbid 625 and notbii 627. (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 128 . . 3 ((𝜑𝜓) → (𝜓𝜑))
21con3d 594 . 2 ((𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))
3 bi1 116 . . 3 ((𝜑𝜓) → (𝜑𝜓))
43con3d 594 . 2 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
52, 4impbid 127 1 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bj-notbii  10984  bj-notbid  10985  bj-dcbi  10986
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