Theorem 19.37vv

Description: Theorem *11.46 in WhiteheadRussell p. 164. Theorem 19.37 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		19.37v	$$⊢\exists y\left(\psi \to \phi \right)\leftrightarrow \left(\psi \to \exists y\phi \right)$$
2	1	exbii	$$⊢\exists x\exists y\left(\psi \to \phi \right)\leftrightarrow \exists x\left(\psi \to \exists y\phi \right)$$
3		19.37v	$$⊢\exists x\left(\psi \to \exists y\phi \right)\leftrightarrow \left(\psi \to \exists x\exists y\phi \right)$$
4	2 3	bitri	$$⊢\exists x\exists y\left(\psi \to \phi \right)\leftrightarrow \left(\psi \to \exists x\exists y\phi \right)$$