Theorem dveeq2-o

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 using ax-c15 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dveeq2-o	$$⊢\neg \forall xx=y\to \left(z=y\to \forall xz=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢z=w\to \forall xz=w$$
2		ax-5	$$⊢z=y\to \forall wz=y$$
3		equequ2	$$⊢w=y\to \left(z=w\leftrightarrow z=y\right)$$
4	1 2 3	dvelimf-o	$$⊢\neg \forall xx=y\to \left(z=y\to \forall xz=y\right)$$