Definition df-nfc 2753

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 1710 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
df-nfc	|- ( F/_ x A <-> A. y F/ x y e. 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 2751	. 2 wff F/_ x A
4		vy	. . . . . 6 setvar y
5	4	cv 1482	. . . . 5 class y
6	5, 2	wcel 1990	. . . 4 wff y e. A
7	6, 1	wnf 1708	. . 3 wff F/ x y e. A
8	7, 4	wal 1481	. 2 wff A. y F/ x y e. A
9	3, 8	wb 196	1 wff ( F/_ x A <-> A. y F/ x y e. A )

This definition is referenced by:  nfci  2754  nfcr  2756  nfcd  2759  nfceqdf  2760  nfnfc1  2767  nfnfc  2774  nfnfcALT  2775  drnfc1  2782  drnfc2  2783  dfnfc2  4454  dfnfc2OLD  4455  nfnid  4897  bj-nfnfc  32853  bj-nfcf  32920