Theorem 19.23vv

Description: Theorem 19.23v extended to two variables. (Contributed by NM, 10-Aug-2004)

	Ref	Expression
Assertion	19.23vv	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\exists y\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.23v	$$⊢\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists y\phi \to \psi \right)$$
2	1	albii	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\exists y\phi \to \psi \right)$$
3		19.23v	$$⊢\forall x\left(\exists y\phi \to \psi \right)\leftrightarrow \left(\exists x\exists y\phi \to \psi \right)$$
4	2 3	bitri	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\exists y\phi \to \psi \right)$$