Theorem orbi2d

Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	bid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	orbi2d	$$⊢\phi \to \left(\left(\theta \vee \psi \right)\leftrightarrow \left(\theta \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	imbi2d	$$⊢\phi \to \left(\left(\neg \theta \to \psi \right)\leftrightarrow \left(\neg \theta \to \chi \right)\right)$$
3		df-or	$$⊢\left(\theta \vee \psi \right)\leftrightarrow \left(\neg \theta \to \psi \right)$$
4		df-or	$$⊢\left(\theta \vee \chi \right)\leftrightarrow \left(\neg \theta \to \chi \right)$$
5	2 3 4	3bitr4g	$$⊢\phi \to \left(\left(\theta \vee \psi \right)\leftrightarrow \left(\theta \vee \chi \right)\right)$$