Theorem 2exanali

Description: Theorem *11.521 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2exanali	$$⊢\neg \exists x\exists y\left(\phi \wedge \neg \psi \right)\leftrightarrow \forall x\forall y\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2nalexn	$$⊢\neg \forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \exists x\exists y\neg \left(\phi \to \psi \right)$$
2	1	con1bii	$$⊢\neg \exists x\exists y\neg \left(\phi \to \psi \right)\leftrightarrow \forall x\forall y\left(\phi \to \psi \right)$$
3		annim	$$⊢\left(\phi \wedge \neg \psi \right)\leftrightarrow \neg \left(\phi \to \psi \right)$$
4	3	2exbii	$$⊢\exists x\exists y\left(\phi \wedge \neg \psi \right)\leftrightarrow \exists x\exists y\neg \left(\phi \to \psi \right)$$
5	2 4	xchnxbir	$$⊢\neg \exists x\exists y\left(\phi \wedge \neg \psi \right)\leftrightarrow \forall x\forall y\left(\phi \to \psi \right)$$