Theorem 3orbi123d

Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994)

	Ref	Expression
Hypotheses	bi3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bi3d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
	bi3d.3	$$⊢\phi \to \left(\eta \leftrightarrow \zeta \right)$$
Assertion	3orbi123d	$$⊢\phi \to \left(\left(\psi \vee \theta \vee \eta \right)\leftrightarrow \left(\chi \vee \tau \vee \zeta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bi3d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3		bi3d.3	$$⊢\phi \to \left(\eta \leftrightarrow \zeta \right)$$
4	1 2	orbi12d	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\leftrightarrow \left(\chi \vee \tau \right)\right)$$
5	4 3	orbi12d	$$⊢\phi \to \left(\left(\left(\psi \vee \theta \right)\vee \eta \right)\leftrightarrow \left(\left(\chi \vee \tau \right)\vee \zeta \right)\right)$$
6		df-3or	$$⊢\left(\psi \vee \theta \vee \eta \right)\leftrightarrow \left(\left(\psi \vee \theta \right)\vee \eta \right)$$
7		df-3or	$$⊢\left(\chi \vee \tau \vee \zeta \right)\leftrightarrow \left(\left(\chi \vee \tau \right)\vee \zeta \right)$$
8	5 6 7	3bitr4g	$$⊢\phi \to \left(\left(\psi \vee \theta \vee \eta \right)\leftrightarrow \left(\chi \vee \tau \vee \zeta \right)\right)$$