Theorem jaoi3 1011

Description: Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)

Hypotheses

Ref	Expression
jaoi3.1	⊢ (𝜑 → 𝜓)
jaoi3.2	⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
jaoi3	⊢ ((𝜑 ∨ 𝜒) → 𝜓)

Proof of Theorem jaoi3

Step	Hyp	Ref	Expression
1		jaoi3.1	. . 3 ⊢ (𝜑 → 𝜓)
2		jaoi3.2	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓)
3	1, 2	jaoi 394	. 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜓)
4	3	jaoi2 1010	1 ⊢ ((𝜑 ∨ 𝜒) → 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:  2mpt20  6882  bropopvvv  7255  bropfvvvv  7257  ssnn0fi  12784  swrdnd  13432