Theorem axc5c711toc7

Description: Rederivation of ax-c7 from axc5c711 . Note that ax-c7 and ax-11 are not used by the rederivation. The use of alimi (which uses ax-c5 ) is allowed since we have already proved axc5c711toc5 . (Contributed by NM, 19-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc5c711toc7	$$⊢\neg \forall x\neg \forall x\phi \to \phi$$

Proof

Step	Hyp	Ref	Expression
1		hba1-o	$$⊢\forall x\phi \to \forall x\forall x\phi$$
2	1	con3i	$$⊢\neg \forall x\forall x\phi \to \neg \forall x\phi$$
3	2	alimi	$$⊢\forall x\neg \forall x\forall x\phi \to \forall x\neg \forall x\phi$$
4	3	sps-o	$$⊢\forall x\forall x\neg \forall x\forall x\phi \to \forall x\neg \forall x\phi$$
5	4	con3i	$$⊢\neg \forall x\neg \forall x\phi \to \neg \forall x\forall x\neg \forall x\forall x\phi$$
6		pm2.21	$$⊢\neg \forall x\forall x\neg \forall x\forall x\phi \to \left(\forall x\forall x\neg \forall x\forall x\phi \to \forall x\phi \right)$$
7		axc5c711	$$⊢\left(\forall x\forall x\neg \forall x\forall x\phi \to \forall x\phi \right)\to \phi$$
8	5 6 7	3syl	$$⊢\neg \forall x\neg \forall x\phi \to \phi$$