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	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )
	ex-natded5.13.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
	ex-natded5.13.3	⊢ ( 𝜑 → ( ¬ 𝜏 → ¬ 𝜒 ) )
Assertion	ex-natded5.13-2	⊢ ( 𝜑 → ( 𝜃 ∨ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )
2		ex-natded5.13.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3		ex-natded5.13.3	⊢ ( 𝜑 → ( ¬ 𝜏 → ¬ 𝜒 ) )
4	3	con4d	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
5	2 4	orim12d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜒 ) → ( 𝜃 ∨ 𝜏 ) ) )
6	1 5	mpd	⊢ ( 𝜑 → ( 𝜃 ∨ 𝜏 ) )