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Metamath Proof Explorer

Theorem sylanbrc

Description: Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009)

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

Proof

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