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Theorem bj-nnor 12780
Description: Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
bj-nnor (¬ ¬ (𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))

Proof of Theorem bj-nnor
StepHypRef Expression
1 ioran 724 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21notbii 640 . 2 (¬ ¬ (𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))
3 imnan 662 . 2 ((¬ 𝜑 → ¬ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))
42, 3bitr4i 186 1 (¬ ¬ (𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bj-nndcALT  12797
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