Theorem nfcvf

Description: If x and y are distinct, then x is not free in y . Usage of this theorem is discouraged because it depends on ax-13 . See nfcv for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-ext . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎw\neg \forall xx=y$$
2		nfv	$$⊢Ⅎxw\in z$$
3		elequ2	$$⊢z=y\to \left(w\in z\leftrightarrow w\in y\right)$$
4	2 3	dvelimnf	$$⊢\neg \forall xx=y\to Ⅎxw\in y$$
5	1 4	nfcd	$$⊢\neg \forall xx=y\to \underset{\_}{Ⅎ}xy$$