Theorem ex-natded5.8-2

Description: A more efficient proof of Theorem 5.8 of Clemente p. 20. For a longer line-by-line translation, see ex-natded5.8 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ex-natded5.8.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → ¬ 𝜃 ) )
	ex-natded5.8.2	⊢ ( 𝜑 → ( 𝜏 → 𝜃 ) )
	ex-natded5.8.3	⊢ ( 𝜑 → 𝜒 )
	ex-natded5.8.4	⊢ ( 𝜑 → 𝜏 )
Assertion	ex-natded5.8-2	⊢ ( 𝜑 → ¬ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.8.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → ¬ 𝜃 ) )
2		ex-natded5.8.2	⊢ ( 𝜑 → ( 𝜏 → 𝜃 ) )
3		ex-natded5.8.3	⊢ ( 𝜑 → 𝜒 )
4		ex-natded5.8.4	⊢ ( 𝜑 → 𝜏 )
5	4 2	mpd	⊢ ( 𝜑 → 𝜃 )
6	3 1	mpan2d	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝜃 ) )
7	5 6	mt2d	⊢ ( 𝜑 → ¬ 𝜓 )