Theorem ex-natded5.2-2

Description: A more efficient proof of Theorem 5.2 of Clemente p. 15. Compare with ex-natded5.2 and ex-natded5.2i . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ex-natded5.2.1	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \theta \right)$$
	ex-natded5.2.2	$$⊢\phi \to \left(\chi \to \psi \right)$$
	ex-natded5.2.3	$$⊢\phi \to \chi$$
Assertion	ex-natded5.2-2	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.2.1	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \theta \right)$$
2		ex-natded5.2.2	$$⊢\phi \to \left(\chi \to \psi \right)$$
3		ex-natded5.2.3	$$⊢\phi \to \chi$$
4	3 2	mpd	$$⊢\phi \to \psi$$
5	4 3 1	mp2and	$$⊢\phi \to \theta$$