This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sylanblrc

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

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

Proof

Step Hyp Ref Expression
1 sylanblrc.1 ⊢ ( 𝜑 → 𝜓 )
2 sylanblrc.2 ⊢ 𝜒
3 sylanblrc.3 ⊢ ( 𝜃 ↔ ( 𝜓 ∧ 𝜒 ) )
4 2 a1i ⊢ ( 𝜑 → 𝜒 )
5 1 4 3 sylanbrc ⊢ ( 𝜑 → 𝜃 )