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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