Theorem notornotel1

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

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

Proof

Step	Hyp	Ref	Expression
1		notornotel1.1	⊢ ( 𝜑 → ¬ ( ¬ 𝜓 ∨ 𝜒 ) )
2		ioran	⊢ ( ¬ ( ¬ 𝜓 ∨ 𝜒 ) ↔ ( ¬ ¬ 𝜓 ∧ ¬ 𝜒 ) )
3	2	biimpi	⊢ ( ¬ ( ¬ 𝜓 ∨ 𝜒 ) → ( ¬ ¬ 𝜓 ∧ ¬ 𝜒 ) )
4		simpl	⊢ ( ( ¬ ¬ 𝜓 ∧ ¬ 𝜒 ) → ¬ ¬ 𝜓 )
5		notnotr	⊢ ( ¬ ¬ 𝜓 → 𝜓 )
6	3 4 5	3syl	⊢ ( ¬ ( ¬ 𝜓 ∨ 𝜒 ) → 𝜓 )
7	1 6	syl	⊢ ( 𝜑 → 𝜓 )