Theorem ex-natded5.13-2

Description: A more efficient proof of Theorem 5.13 of Clemente p. 20. Compare with ex-natded5.13 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ex-natded5.13.1	$$⊢\phi \to \left(\psi \vee \chi \right)$$
	ex-natded5.13.2	$$⊢\phi \to \left(\psi \to \theta \right)$$
	ex-natded5.13.3	$$⊢\phi \to \left(\neg \tau \to \neg \chi \right)$$
Assertion	ex-natded5.13-2	$$⊢\phi \to \left(\theta \vee \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	$$⊢\phi \to \left(\psi \vee \chi \right)$$
2		ex-natded5.13.2	$$⊢\phi \to \left(\psi \to \theta \right)$$
3		ex-natded5.13.3	$$⊢\phi \to \left(\neg \tau \to \neg \chi \right)$$
4	3	con4d	$$⊢\phi \to \left(\chi \to \tau \right)$$
5	2 4	orim12d	$$⊢\phi \to \left(\left(\psi \vee \chi \right)\to \left(\theta \vee \tau \right)\right)$$
6	1 5	mpd	$$⊢\phi \to \left(\theta \vee \tau \right)$$