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Metamath Proof Explorer

Theorem sylc

Description: A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994) (Revised by NM, 13-Jul-2013)

Ref Expression
Hypotheses sylc.1 ⊢ ( 𝜑 → 𝜓 )
sylc.2 ⊢ ( 𝜑 → 𝜒 )
sylc.3 ⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
Assertion sylc ( 𝜑 → 𝜃 )

Proof

Step Hyp Ref Expression
1 sylc.1 ⊢ ( 𝜑 → 𝜓 )
2 sylc.2 ⊢ ( 𝜑 → 𝜒 )
3 sylc.3 ⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
4 1 2 3 syl2im ⊢ ( 𝜑 → ( 𝜑 → 𝜃 ) )
5 4 pm2.43i ⊢ ( 𝜑 → 𝜃 )