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Definition df-nf 1324
Description: Define the not-free predicate for wffs. This is read "x is not free in φ". Not-free means that the value of x cannot affect the value of φ, e.g., any occurrence of x in φ 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 1634). An example of where this is used is stdpc5 1450. See nf2 1532 for an alternative definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, 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 not-free notion 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 only considers syntactic freedom). For example, x is effectively not free in the bare expression x = x, even though x would be considered free in the usual textbook definition, because the value of x in the expression x = x cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf (Ⅎxφx(φxφ))

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 setvar x
31, 2wnf 1323 . 2 wff xφ
41, 2wal 1222 . . . 4 wff xφ
51, 4wi 4 . . 3 wff (φxφ)
65, 2wal 1222 . 2 wff x(φxφ)
73, 6wb 98 1 wff (Ⅎxφx(φxφ))
Colors of variables: wff set class
This definition is referenced by:  nfi  1325  nfbii  1336  nfr  1385  nfd  1390  nfbidf  1406  nfnf1  1410  nford  1433  nfand  1434  nfnf  1443  nfalt  1444  19.21t  1448  nfimd  1451  19.9t  1507  nfnt  1520  nf2  1532  drnf1  1595  drnf2  1596  nfexd  1618  dveeq2or  1671  nfsb2or  1692  nfdv  1731  nfsbxy  1792  nfsbxyt  1793  sbcomxyyz  1820  sbnf2  1831  dvelimALT  1860  dvelimfv  1861  nfsb4t  1864  dvelimor  1868  oprabidlem  5449  bj-nfalt  8241
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