Theorem jaao

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

	Ref	Expression
Hypotheses	jaao.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	jaao.2	$$⊢\theta \to \left(\tau \to \chi \right)$$
Assertion	jaao	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \vee \tau \right)\to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		jaao.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		jaao.2	$$⊢\theta \to \left(\tau \to \chi \right)$$
3	1	adantr	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \to \chi \right)$$
4	2	adantl	$$⊢\left(\phi \wedge \theta \right)\to \left(\tau \to \chi \right)$$
5	3 4	jaod	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \vee \tau \right)\to \chi \right)$$