Theorem r19.36zv

Description: Restricted quantifier version of Theorem 19.36 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Assertion	r19.36zv	$$⊢A\ne \varnothing \to \left(\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.35	$$⊢\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)$$
2		r19.9rzv	$$⊢A\ne \varnothing \to \left(\psi \leftrightarrow \exists x\in A\psi \right)$$
3	2	imbi2d	$$⊢A\ne \varnothing \to \left(\left(\forall x\in A\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \exists x\in A\psi \right)\right)$$
4	1 3	bitr4id	$$⊢A\ne \varnothing \to \left(\exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\phi \to \psi \right)\right)$$