Theorem 3orbi123i

Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994)

	Ref	Expression
Hypotheses	bi3.1	$$⊢\phi \leftrightarrow \psi$$
	bi3.2	$$⊢\chi \leftrightarrow \theta$$
	bi3.3	$$⊢\tau \leftrightarrow \eta$$
Assertion	3orbi123i	$$⊢\left(\phi \vee \chi \vee \tau \right)\leftrightarrow \left(\psi \vee \theta \vee \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		bi3.1	$$⊢\phi \leftrightarrow \psi$$
2		bi3.2	$$⊢\chi \leftrightarrow \theta$$
3		bi3.3	$$⊢\tau \leftrightarrow \eta$$
4	1 2	orbi12i	$$⊢\left(\phi \vee \chi \right)\leftrightarrow \left(\psi \vee \theta \right)$$
5	4 3	orbi12i	$$⊢\left(\left(\phi \vee \chi \right)\vee \tau \right)\leftrightarrow \left(\left(\psi \vee \theta \right)\vee \eta \right)$$
6		df-3or	$$⊢\left(\phi \vee \chi \vee \tau \right)\leftrightarrow \left(\left(\phi \vee \chi \right)\vee \tau \right)$$
7		df-3or	$$⊢\left(\psi \vee \theta \vee \eta \right)\leftrightarrow \left(\left(\psi \vee \theta \right)\vee \eta \right)$$
8	5 6 7	3bitr4i	$$⊢\left(\phi \vee \chi \vee \tau \right)\leftrightarrow \left(\psi \vee \theta \vee \eta \right)$$