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Theorem bj-nnan 13053
Description: The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnan (¬ ¬ (𝜑𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

Proof of Theorem bj-nnan
StepHypRef Expression
1 simpl 108 . . . 4 ((𝜑𝜓) → 𝜑)
21con3i 621 . . 3 𝜑 → ¬ (𝜑𝜓))
32con3i 621 . 2 (¬ ¬ (𝜑𝜓) → ¬ ¬ 𝜑)
4 simpr 109 . . . 4 ((𝜑𝜓) → 𝜓)
54con3i 621 . . 3 𝜓 → ¬ (𝜑𝜓))
65con3i 621 . 2 (¬ ¬ (𝜑𝜓) → ¬ ¬ 𝜓)
73, 6jca 304 1 (¬ ¬ (𝜑𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103
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 603  ax-in2 604
This theorem is referenced by:  bj-stan  13060  bj-stand  13061
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