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	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
	ex-natded5.2.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
	ex-natded5.2.3	⊢ ( 𝜑 → 𝜒 )
Assertion	ex-natded5.2-2	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.2.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
2		ex-natded5.2.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
3		ex-natded5.2.3	⊢ ( 𝜑 → 𝜒 )
4	3 2	mpd	⊢ ( 𝜑 → 𝜓 )
5	4 3 1	mp2and	⊢ ( 𝜑 → 𝜃 )