Theorem axc711

Description: Proof of a single axiom that can replace both ax-c7 and ax-11 . See axc711toc7 and axc711to11 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc711	$$⊢\neg \forall x\neg \forall y\forall x\phi \to \forall y\phi$$

Proof

Step	Hyp	Ref	Expression
1		ax-11	$$⊢\forall y\forall x\phi \to \forall x\forall y\phi$$
2	1	con3i	$$⊢\neg \forall x\forall y\phi \to \neg \forall y\forall x\phi$$
3	2	alimi	$$⊢\forall x\neg \forall x\forall y\phi \to \forall x\neg \forall y\forall x\phi$$
4	3	con3i	$$⊢\neg \forall x\neg \forall y\forall x\phi \to \neg \forall x\neg \forall x\forall y\phi$$
5		ax-c7	$$⊢\neg \forall x\neg \forall x\forall y\phi \to \forall y\phi$$
6	4 5	syl	$$⊢\neg \forall x\neg \forall y\forall x\phi \to \forall y\phi$$