Theorem jaodan

Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005)

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

Proof

Step	Hyp	Ref	Expression
1		jaodan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		jaodan.2	$$⊢\left(\phi \wedge \theta \right)\to \chi$$
3	1	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
4	2	ex	$$⊢\phi \to \left(\theta \to \chi \right)$$
5	3 4	jaod	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \chi \right)$$
6	5	imp	$$⊢\left(\phi \wedge \left(\psi \vee \theta \right)\right)\to \chi$$