Theorem jaoa

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

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

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	adantrd	$$⊢\phi \to \left(\left(\psi \wedge \tau \right)\to \chi \right)$$
4	2	adantld	$$⊢\theta \to \left(\left(\psi \wedge \tau \right)\to \chi \right)$$
5	3 4	jaoi	$$⊢\left(\phi \vee \theta \right)\to \left(\left(\psi \wedge \tau \right)\to \chi \right)$$