Theorem dveeq1-o

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 using ax-c11 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dveeq1-o	$$⊢\neg \forall xx=y\to \left(y=z\to \forall xy=z\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢w=z\to \forall xw=z$$
2		ax-5	$$⊢y=z\to \forall wy=z$$
3		equequ1	$$⊢w=y\to \left(w=z\leftrightarrow y=z\right)$$
4	1 2 3	dvelimf-o	$$⊢\neg \forall xx=y\to \left(y=z\to \forall xy=z\right)$$