Theorem syl3anbrc

Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014)

	Ref	Expression
Hypotheses	syl3anbrc.1	⊢ ( 𝜑 → 𝜓 )
	syl3anbrc.2	⊢ ( 𝜑 → 𝜒 )
	syl3anbrc.3	⊢ ( 𝜑 → 𝜃 )
	syl3anbrc.4	⊢ ( 𝜏 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
Assertion	syl3anbrc	⊢ ( 𝜑 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl3anbrc.1	⊢ ( 𝜑 → 𝜓 )
2		syl3anbrc.2	⊢ ( 𝜑 → 𝜒 )
3		syl3anbrc.3	⊢ ( 𝜑 → 𝜃 )
4		syl3anbrc.4	⊢ ( 𝜏 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
5	1 2 3	3jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
6	5 4	sylibr	⊢ ( 𝜑 → 𝜏 )