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Metamath Proof Explorer

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 )