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Metamath Proof Explorer

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 )