Theorem notornotel1 38785

Description: A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)

Hypothesis

Ref	Expression
notornotel1.1	⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒))

Assertion

Ref	Expression
notornotel1	⊢ (𝜑 → 𝜓)

Proof of Theorem notornotel1

Step	Hyp	Ref	Expression
1		notornotel1.1	. 2 ⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒))
2		ioran 999	. . 3 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ↔ (¬ ¬ 𝜓 ∧ ¬ 𝜒))
3	2	biimpi 219	. 2 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) → (¬ ¬ 𝜓 ∧ ¬ 𝜒))
4		simpl 488	. 2 ⊢ ((¬ ¬ 𝜓 ∧ ¬ 𝜒) → ¬ ¬ 𝜓)
5		notnotr 131	. 2 ⊢ (¬ ¬ 𝜓 → 𝜓)
6	1, 3, 4, 5	4syl 20	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  notornotel2  38786  ac6s6  38862