Theorem 3jaao

Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Andrew Salmon, 13-May-2011) (Proof shortened by Garrett Katz, 16-Jun-2026)

	Ref	Expression
Hypotheses	3jaao.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	3jaao.2	⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
	3jaao.3	⊢ ( 𝜂 → ( 𝜁 → 𝜒 ) )
Assertion	3jaao	⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜂 ) → ( ( 𝜓 ∨ 𝜏 ∨ 𝜁 ) → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		3jaao.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		3jaao.2	⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
3		3jaao.3	⊢ ( 𝜂 → ( 𝜁 → 𝜒 ) )
4		3jao	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜒 ) ∧ ( 𝜁 → 𝜒 ) ) → ( ( 𝜓 ∨ 𝜏 ∨ 𝜁 ) → 𝜒 ) )
5	1 2 3 4	syl3an	⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜂 ) → ( ( 𝜓 ∨ 𝜏 ∨ 𝜁 ) → 𝜒 ) )