Theorem jaoded 45508

Description: Deduction form of jao 975. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
jaoded.1	⊢ (𝜑 → (𝜓 → 𝜒))
jaoded.2	⊢ (𝜃 → (𝜏 → 𝜒))
jaoded.3	⊢ (𝜂 → (𝜓 ∨ 𝜏))

Assertion

Ref	Expression
jaoded	⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → 𝜒)

Proof of Theorem jaoded

Step	Hyp	Ref	Expression
1		jaoded.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		jaoded.2	. 2 ⊢ (𝜃 → (𝜏 → 𝜒))
3		jaoded.3	. 2 ⊢ (𝜂 → (𝜓 ∨ 𝜏))
4		jao 975	. . 3 ⊢ ((𝜓 → 𝜒) → ((𝜏 → 𝜒) → ((𝜓 ∨ 𝜏) → 𝜒)))
5	4	3imp 1128	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜒) ∧ (𝜓 ∨ 𝜏)) → 𝜒)
6	1, 2, 3, 5	syl3an 1178	1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∧ w3a 1103

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  df-3an 1105

This theorem is used by:  suctrALT3  45865