Definition df-nfc 2381

Description: Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1514 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
df-nfc	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Detailed syntax breakdown of Definition df-nfc

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		cA	. . 3 class 𝐴
3	1, 2	wnfc 2379	. 2 wff Ⅎ𝑥𝐴
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1401	. . . . 5 class 𝑦
6	5, 2	wcel 2209	. . . 4 wff 𝑦 ∈ 𝐴
7	6, 1	wnf 1513	. . 3 wff Ⅎ𝑥 𝑦 ∈ 𝐴
8	7, 4	wal 1400	. 2 wff ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴
9	3, 8	wb 105	1 wff (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)

This definition is referenced by:  nfci  2382  nfcr  2384  nfcd  2387  nfceqi  2388  nfceqdf  2391  nfnfc1  2395  nfnfc  2399  drnfc1  2409  drnfc2  2410  dfnfc2  3951