Theorem 19.37

Description: Theorem 19.37 of Margaris p. 90. See 19.37v for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	19.37.1	$$⊢Ⅎx\phi$$
Assertion	19.37	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.37.1	$$⊢Ⅎx\phi$$
2		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
3	1	19.3	$$⊢\forall x\phi \leftrightarrow \phi$$
4	3	imbi1i	$$⊢\left(\forall x\phi \to \exists x\psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$
5	2 4	bitri	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \exists x\psi \right)$$