Theorem orsild 1019

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 999	. . 3 ⊢ (¬ (𝜓 ∨ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (¬ 𝜓 ∧ ¬ 𝜒))
4	3	simpld 500	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:  angmgmaddeu1  29313  angmgmaddcpbl  29324  symquadprlng  29374  prlngsymquadlem  29375  prlngsymquadopp  29377  quadcgrprlng  29378