Theorem orbi12d

Description: Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypotheses	bi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	orbi12d	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\leftrightarrow \left(\chi \vee \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3	1	orbi1d	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\leftrightarrow \left(\chi \vee \theta \right)\right)$$
4	2	orbi2d	$$⊢\phi \to \left(\left(\chi \vee \theta \right)\leftrightarrow \left(\chi \vee \tau \right)\right)$$
5	3 4	bitrd	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\leftrightarrow \left(\chi \vee \tau \right)\right)$$