Definition df-nfc 2164

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 1347 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 2162	. 2 wff ℲxA
4		vy	. . . . . 6 setvar y
5	4	cv 1241	. . . . 5 class y
6	5, 2	wcel 1390	. . . 4 wff y ∈ A
7	6, 1	wnf 1346	. . 3 wff Ⅎx y ∈ A
8	7, 4	wal 1240	. 2 wff ∀yℲx y ∈ A
9	3, 8	wb 98	1 wff (ℲxA ↔ ∀yℲx y ∈ A)

This definition is referenced by:  nfci  2165  nfcr  2167  nfcd  2170  nfceqi  2171  nfceqdf  2174  nfnfc1  2178  nfnfc  2181  drnfc1  2191  drnfc2  2192  dfnfc2  3589