Theorem axc5c711to11

Description: Rederivation of ax-11 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	axc5c711to11	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		axc5c711toc7	$$⊢\neg \forall y\neg \forall y\neg \forall x\forall y\phi \to \neg \forall x\forall y\phi$$
2	1	con4i	$$⊢\forall x\forall y\phi \to \forall y\neg \forall y\neg \forall x\forall y\phi$$
3		pm2.21	$$⊢\neg \forall x\forall y\neg \forall x\forall y\phi \to \left(\forall x\forall y\neg \forall x\forall y\phi \to \forall x\phi \right)$$
4		axc5c711	$$⊢\left(\forall x\forall y\neg \forall x\forall y\phi \to \forall x\phi \right)\to \phi$$
5	3 4	syl	$$⊢\neg \forall x\forall y\neg \forall x\forall y\phi \to \phi$$
6	5	alimi	$$⊢\forall x\neg \forall x\forall y\neg \forall x\forall y\phi \to \forall x\phi$$
7		axc5c711toc7	$$⊢\neg \forall x\neg \forall x\forall y\neg \forall x\forall y\phi \to \forall y\neg \forall x\forall y\phi$$
8	6 7	nsyl4	$$⊢\neg \forall y\neg \forall x\forall y\phi \to \forall x\phi$$
9	8	alimi	$$⊢\forall y\neg \forall y\neg \forall x\forall y\phi \to \forall y\forall x\phi$$
10	2 9	syl	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$