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 )