Theorem nfcvb

Description: The "distinctor" expression -. A. x x = y , stating that x and y are not the same variable, can be written in terms of F/ in the obvious way. This theorem is not true in a one-element domain, because then F/_ x y and A. x x = y will both be true. (Contributed by Mario Carneiro, 8-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . (New usage is discouraged.)

	Ref	Expression
Assertion	nfcvb	$$⊢\underset{\_}{Ⅎ}xy\leftrightarrow \neg \forall xx=y$$

Proof

Step	Hyp	Ref	Expression
1		nfnid	$$⊢\neg \underset{\_}{Ⅎ}yy$$
2		eqidd	$$⊢\forall xx=y\to y=y$$
3	2	drnfc1	$$⊢\forall xx=y\to \left(\underset{\_}{Ⅎ}xy\leftrightarrow \underset{\_}{Ⅎ}yy\right)$$
4	1 3	mtbiri	$$⊢\forall xx=y\to \neg \underset{\_}{Ⅎ}xy$$
5	4	con2i	$$⊢\underset{\_}{Ⅎ}xy\to \neg \forall xx=y$$
6		nfcvf	$$⊢\neg \forall xx=y\to \underset{\_}{Ⅎ}xy$$
7	5 6	impbii	$$⊢\underset{\_}{Ⅎ}xy\leftrightarrow \neg \forall xx=y$$