Theorem orsild 33059

Description: A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)

Hypothesis

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

Assertion

Ref	Expression
orsild	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem orsild

Step	Hyp	Ref	Expression
1		orsild.1	. . 3 ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))
2		ioran 510	. . 3 ⊢ (¬ (𝜓 ∨ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
3	1, 2	sylib 207	. 2 ⊢ (𝜑 → (¬ 𝜓 ∧ ¬ 𝜒))
4	3	simpld 474	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by: (None)