Theorem nfeqf2

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 9-Jun-2019) Remove dependency on ax-12 . (Revised by Wolf Lammen, 16-Dec-2022) (New usage is discouraged.)

	Ref	Expression
Assertion	nfeqf2	$$⊢\neg \forall xx=y\to Ⅎxz=y$$

Proof

Step	Hyp	Ref	Expression
1		exnal	$$⊢\exists x\neg x=y\leftrightarrow \neg \forall xx=y$$
2		hbe1	$$⊢\exists xz=y\to \forall x\exists xz=y$$
3		ax13lem2	$$⊢\neg x=y\to \left(\exists xz=y\to z=y\right)$$
4		ax13lem1	$$⊢\neg x=y\to \left(z=y\to \forall xz=y\right)$$
5	3 4	syldc	$$⊢\exists xz=y\to \left(\neg x=y\to \forall xz=y\right)$$
6	2 5	eximdh	$$⊢\exists xz=y\to \left(\exists x\neg x=y\to \exists x\forall xz=y\right)$$
7		hbe1a	$$⊢\exists x\forall xz=y\to \forall xz=y$$
8	6 7	syl6com	$$⊢\exists x\neg x=y\to \left(\exists xz=y\to \forall xz=y\right)$$
9	8	nfd	$$⊢\exists x\neg x=y\to Ⅎxz=y$$
10	1 9	sylbir	$$⊢\neg \forall xx=y\to Ⅎxz=y$$