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Metamath Proof Explorer

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	\|- ( F/_ x y <-> -. A. x x = y )

Proof

Step	Hyp	Ref	Expression
1		nfnid	\|- -. F/_ y y
2		eqidd	\|- ( A. x x = y -> y = y )
3	2	drnfc1	\|- ( A. x x = y -> ( F/_ x y <-> F/_ y y ) )
4	1 3	mtbiri	\|- ( A. x x = y -> -. F/_ x y )
5	4	con2i	\|- ( F/_ x y -> -. A. x x = y )
6		nfcvf	\|- ( -. A. x x = y -> F/_ x y )
7	5 6	impbii	\|- ( F/_ x y <-> -. A. x x = y )