Theorem r19.45zv

Description: Restricted version of Theorem 19.45 of Margaris p. 90. (Contributed by NM, 27-May-1998)

	Ref	Expression
Assertion	r19.45zv	$$⊢A\ne \varnothing \to \left(\exists x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\in A\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.43	$$⊢\exists x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\in A\phi \vee \exists x\in A\psi \right)$$
2		r19.9rzv	$$⊢A\ne \varnothing \to \left(\phi \leftrightarrow \exists x\in A\phi \right)$$
3	2	orbi1d	$$⊢A\ne \varnothing \to \left(\left(\phi \vee \exists x\in A\psi \right)\leftrightarrow \left(\exists x\in A\phi \vee \exists x\in A\psi \right)\right)$$
4	1 3	bitr4id	$$⊢A\ne \varnothing \to \left(\exists x\in A\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\in A\psi \right)\right)$$