Theorem jaao

Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999)

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

Proof

Step	Hyp	Ref	Expression
1		jaao.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		jaao.2	⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
3	1	adantr	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → 𝜒 ) )
4	2	adantl	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜏 → 𝜒 ) )
5	3 4	jaod	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∨ 𝜏 ) → 𝜒 ) )