Theorem ax6fromc10

Description: Rederivation of axiom ax-6 from ax-c7 , ax-c10 , ax-gen and propositional calculus. See axc10 for the derivation of ax-c10 from ax-6 . Lemma L18 in Megill p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993) (Proof modification is discouraged.) Use ax-6 instead. (New usage is discouraged.)

	Ref	Expression
Assertion	ax6fromc10	$$⊢\neg \forall x\neg x=y$$

Proof

Step	Hyp	Ref	Expression
1		ax-c10	$$⊢\forall x\left(x=y\to \forall x\neg \forall x\neg x=y\right)\to \neg \forall x\neg x=y$$
2		ax-c7	$$⊢\neg \forall x\neg \forall x\neg x=y\to \neg x=y$$
3	2	con4i	$$⊢x=y\to \forall x\neg \forall x\neg x=y$$
4	1 3	mpg	$$⊢\neg \forall x\neg x=y$$