Theorem nfcri

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016) Avoid ax-10 , ax-11 . (Revised by GG, 23-May-2024) Avoid ax-12 (adopting Wolf Lammen's 13-May-2023 proof). (Revised by SN, 3-Jun-2024)

	Ref	Expression
Hypothesis	nfcri.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	nfcri	$$⊢Ⅎxy\in A$$

Proof

Step	Hyp	Ref	Expression
1		nfcri.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfcr	$$⊢\underset{\_}{Ⅎ}xA\to Ⅎxy\in A$$
3	1 2	ax-mp	$$⊢Ⅎxy\in A$$