Theorem axc711toc7

Description: Rederivation of ax-c7 from axc711 . Note that ax-c7 and ax-11 are not used by the rederivation. (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc711toc7	$$⊢\neg \forall x\neg \forall x\phi \to \phi$$

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	con3i	$$⊢\neg \forall x\neg \forall x\phi \to \neg \forall x\neg \forall x\forall x\phi$$
5		axc711	$$⊢\neg \forall x\neg \forall x\forall x\phi \to \forall x\phi$$
6		ax-c5	$$⊢\forall x\phi \to \phi$$
7	4 5 6	3syl	$$⊢\neg \forall x\neg \forall x\phi \to \phi$$