Definition df-nfc 2479

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 "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1545 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 2477	. 2 wff ℲxA
4		vy	. . . . . 6 setvar y
5	4	cv 1641	. . . . 5 class y
6	5, 2	wcel 1710	. . . 4 wff y ∈ A
7	6, 1	wnf 1544	. . 3 wff Ⅎx y ∈ A
8	7, 4	wal 1540	. 2 wff ∀yℲx y ∈ A
9	3, 8	wb 176	1 wff (ℲxA ↔ ∀yℲx y ∈ A)

This definition is referenced by:  nfci  2480  nfcr  2482  nfcd  2485  nfceqi  2486  nfceqdf  2489  nfnfc1  2493  nfnfc  2496  drnfc1  2506  drnfc2  2507  dfnfc2  3910