Theorem 19.36vv

Description: Theorem *11.43 in WhiteheadRussell p. 163. Theorem 19.36 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	19.36vv	$$⊢\exists x\exists y\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\forall y\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.36v	$$⊢\exists y\left(\phi \to \psi \right)\leftrightarrow \left(\forall y\phi \to \psi \right)$$
2	1	exbii	$$⊢\exists x\exists y\left(\phi \to \psi \right)\leftrightarrow \exists x\left(\forall y\phi \to \psi \right)$$
3		19.36v	$$⊢\exists x\left(\forall y\phi \to \psi \right)\leftrightarrow \left(\forall x\forall y\phi \to \psi \right)$$
4	2 3	bitri	$$⊢\exists x\exists y\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\forall y\phi \to \psi \right)$$