Theorem jaoi2 1002

Description: Inference removing a negated conjunct in a disjunction of an antecedent if this conjunct is part of the disjunction. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)

Hypothesis

Ref	Expression
jaoi2.1	⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜓)

Assertion

Ref	Expression
jaoi2	⊢ ((𝜑 ∨ 𝜒) → 𝜓)

Proof of Theorem jaoi2

Step	Hyp	Ref	Expression
1		pm5.63 961	. 2 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜒)))
2		jaoi2.1	. 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜓)
3	1, 2	sylbi 206	1 ⊢ ((𝜑 ∨ 𝜒) → 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:  jaoi3  1003