Theorem bj-nnan 16776

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	⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

Proof of Theorem bj-nnan

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	con3i 641	. . 3 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3	2	con3i 641	. 2 ⊢ (¬ ¬ (𝜑 ∧ 𝜓) → ¬ ¬ 𝜑)
4		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
5	4	con3i 641	. . 3 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
6	5	con3i 641	. 2 ⊢ (¬ ¬ (𝜑 ∧ 𝜓) → ¬ ¬ 𝜓)
7	3, 6	jca 306	1 ⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This theorem is used by:  bj-stan  16787  bj-stand  16788