Theorem 19.37v

Description: Version of 19.37 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	19.37v	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
2		19.3v	$$⊢\forall x\phi \leftrightarrow \phi$$
3	2	imbi1i	$$⊢\left(\forall x\phi \to \exists x\psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$
4	1 3	bitri	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$