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Theorem bj-notbi 10019
Description: Equivalence property for negation. TODO: minimize all theorems using notbid 592 and notbii 594. (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 561 . 2 ((𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))
3 bi1 111 . . 3 ((𝜑𝜓) → (𝜑𝜓))
43con3d 561 . 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  10020  bj-notbid  10021  bj-dcbi  10022
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