Theorem notornotel1 33897

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 511	. . . 4 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ↔ (¬ ¬ 𝜓 ∧ ¬ 𝜒))
3	2	biimpi 206	. . 3 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) → (¬ ¬ 𝜓 ∧ ¬ 𝜒))
4		simpl 473	. . 3 ⊢ ((¬ ¬ 𝜓 ∧ ¬ 𝜒) → ¬ ¬ 𝜓)
5		notnotr 125	. . 3 ⊢ (¬ ¬ 𝜓 → 𝜓)
6	3, 4, 5	3syl 18	. 2 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) → 𝜓)
7	1, 6	syl 17	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by:  notornotel2  33898  ac6s6  33980