Theorem jaoa

Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008)

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

Proof

Step	Hyp	Ref	Expression
1		jaao.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		jaao.2	⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
3	1	adantrd	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )
4	2	adantld	⊢ ( 𝜃 → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )
5	3 4	jaoi	⊢ ( ( 𝜑 ∨ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )