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Metamath Proof Explorer

Theorem sylib

Description: A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993)

Ref Expression
Hypotheses sylib.1 ⊢ φ → ψ
sylib.2 ⊢ ψ ↔ χ
Assertion sylib ⊢ φ → χ

Proof

Step Hyp Ref Expression
1 sylib.1 ⊢ φ → ψ
2 sylib.2 ⊢ ψ ↔ χ
3 2 biimpi ⊢ ψ → χ
4 1 3 syl ⊢ φ → χ