Definition df-nf 1484

Description: Define the not-free predicate for wffs. This is read " x is not free in ph". Not-free means that the value of x cannot affect the value of ph, e.g., any occurrence of x in ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1800). An example of where this is used is stdpc5 1607. See nf2 1691 for an alternate definition which does not involve nested quantifiers on the same variable.

Nonfreeness is a commonly used condition, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the notion of nonfreeness within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for "not free" (which considers syntactic freedom). For example, x is effectively not free in the expression x = x (even though x is syntactically free in it, so would be considered "free" in the usual textbook definition) because the value of x in the formula x = x does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 1725. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
df-nf	|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )

Detailed syntax breakdown of Definition df-nf

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3	1, 2	wnf 1483	. 2 wff F/ x ph
4	1, 2	wal 1371	. . . 4 wff A. x ph
5	1, 4	wi 4	. . 3 wff ( ph -> A. x ph )
6	5, 2	wal 1371	. 2 wff A. x ( ph -> A. x ph )
7	3, 6	wb 105	1 wff ( F/ x ph <-> A. x ( ph -> A. x ph ) )

This definition is referenced by:  nfi  1485  nfbii  1496  nfr  1541  nfd  1546  nfbidf  1562  nfnf1  1567  nford  1590  nfand  1591  nfal  1599  nfnf  1600  nfalt  1601  19.21t  1605  nfimd  1608  19.9t  1665  nfnt  1679  nf2  1691  drnf1  1756  drnf2  1757  nfexd  1784  dveeq2or  1839  nfsb2or  1860  nfdv  1900  nfsbxy  1970  nfsbxyt  1971  sbcomxyyz  2000  sbnf2  2009  dvelimALT  2038  dvelimfv  2039  nfsb4t  2042  dvelimor  2046  oprabidlem  5975  bj-nfalt  15704