Theorem pm11.52

Description: Theorem *11.52 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.52	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \neg \forall x\forall y\left(\phi \to \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-an	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \neg \left(\phi \to \neg \psi \right)$$
2	1	2exbii	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \exists x\exists y\neg \left(\phi \to \neg \psi \right)$$
3		2nalexn	$$⊢\neg \forall x\forall y\left(\phi \to \neg \psi \right)\leftrightarrow \exists x\exists y\neg \left(\phi \to \neg \psi \right)$$
4	2 3	bitr4i	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \neg \forall x\forall y\left(\phi \to \neg \psi \right)$$