Theorem syl3anbrc

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

	Ref	Expression
Hypotheses	syl3anbrc.1	$$⊢\phi \to \psi$$
	syl3anbrc.2	$$⊢\phi \to \chi$$
	syl3anbrc.3	$$⊢\phi \to \theta$$
	syl3anbrc.4	$$⊢\tau \leftrightarrow \left(\psi \wedge \chi \wedge \theta \right)$$
Assertion	syl3anbrc	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syl3anbrc.1	$$⊢\phi \to \psi$$
2		syl3anbrc.2	$$⊢\phi \to \chi$$
3		syl3anbrc.3	$$⊢\phi \to \theta$$
4		syl3anbrc.4	$$⊢\tau \leftrightarrow \left(\psi \wedge \chi \wedge \theta \right)$$
5	1 2 3	3jca	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$
6	5 4	sylibr	$$⊢\phi \to \tau$$