Theorem 19.24

Description: Theorem 19.24 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.24	$$⊢\left(\forall x\phi \to \forall x\psi \right)\to \exists x\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.2	$$⊢\forall x\psi \to \exists x\psi$$
2	1	imim2i	$$⊢\left(\forall x\phi \to \forall x\psi \right)\to \left(\forall x\phi \to \exists x\psi \right)$$
3		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
4	2 3	sylibr	$$⊢\left(\forall x\phi \to \forall x\psi \right)\to \exists x\left(\phi \to \psi \right)$$