Theorem nfcri

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016) Avoid ax-10 , ax-11 . (Revised by Gino Giotto, 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	⊢ Ⅎ 𝑥 𝐴
Assertion	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		nfcri.1	⊢ Ⅎ 𝑥 𝐴
2		nfcr	⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
3	1 2	ax-mp	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴