Theorem syl3anbr

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011)

	Ref	Expression
Hypotheses	syl3anbr.1	$$⊢\psi \leftrightarrow \phi$$
	syl3anbr.2	$$⊢\theta \leftrightarrow \chi$$
	syl3anbr.3	$$⊢\eta \leftrightarrow \tau$$
	syl3anbr.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
Assertion	syl3anbr	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anbr.1	$$⊢\psi \leftrightarrow \phi$$
2		syl3anbr.2	$$⊢\theta \leftrightarrow \chi$$
3		syl3anbr.3	$$⊢\eta \leftrightarrow \tau$$
4		syl3anbr.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
5	1	bicomi	$$⊢\phi \leftrightarrow \psi$$
6	2	bicomi	$$⊢\chi \leftrightarrow \theta$$
7	3	bicomi	$$⊢\tau \leftrightarrow \eta$$
8	5 6 7 4	syl3anb	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$