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Metamath Proof Explorer

Theorem sylbb

Description: A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019)

Ref Expression
Hypotheses sylbb.1 ⊢ ( 𝜑 ↔ 𝜓 )
sylbb.2 ⊢ ( 𝜓 ↔ 𝜒 )
Assertion sylbb ( 𝜑 → 𝜒 )

Proof

Step Hyp Ref Expression
1 sylbb.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 sylbb.2 ⊢ ( 𝜓 ↔ 𝜒 )
3 2 biimpi ⊢ ( 𝜓 → 𝜒 )
4 1 3 sylbi ⊢ ( 𝜑 → 𝜒 )