Theorem orim12d

Description: Disjoin antecedents and consequents in a deduction. See orim12dALT for a proof which does not depend on df-an . (Contributed by NM, 10-May-1994)

	Ref	Expression
Hypotheses	orim12d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	orim12d.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
Assertion	orim12d	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orim12d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		orim12d.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
3		pm3.48	$$⊢\left(\left(\psi \to \chi \right)\wedge \left(\theta \to \tau \right)\right)\to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$