Theorem 2nalexn

Description: Part of theorem *11.5 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2nalexn	$$⊢\neg \forall x\forall y\phi \leftrightarrow \exists x\exists y\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		df-ex	$$⊢\exists x\exists y\neg \phi \leftrightarrow \neg \forall x\neg \exists y\neg \phi$$
2		alex	$$⊢\forall y\phi \leftrightarrow \neg \exists y\neg \phi$$
3	2	albii	$$⊢\forall x\forall y\phi \leftrightarrow \forall x\neg \exists y\neg \phi$$
4	1 3	xchbinxr	$$⊢\exists x\exists y\neg \phi \leftrightarrow \neg \forall x\forall y\phi$$
5	4	bicomi	$$⊢\neg \forall x\forall y\phi \leftrightarrow \exists x\exists y\neg \phi$$