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Metamath Proof Explorer

Theorem sylanblc

Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019)

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

Proof

Step Hyp Ref Expression
1 sylanblc.1 ⊢ ( 𝜑 → 𝜓 )
2 sylanblc.2 ⊢ 𝜒
3 sylanblc.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 )
4 3 biimpi ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
5 1 2 4 sylancl ⊢ ( 𝜑 → 𝜃 )