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Definition df-nf 1347
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 1657). An example of where this is used is stdpc5 1473. See nf2 1555 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 1346 . 2 wff xφ
41, 2wal 1240 . . . 4 wff xφ
51, 4wi 4 . . 3 wff (φxφ)
65, 2wal 1240 . 2 wff x(φxφ)
73, 6wb 98 1 wff (Ⅎxφx(φxφ))
Colors of variables: wff set class
This definition is referenced by:  nfi  1348  nfbii  1359  nfr  1408  nfd  1413  nfbidf  1429  nfnf1  1433  nford  1456  nfand  1457  nfnf  1466  nfalt  1467  19.21t  1471  nfimd  1474  19.9t  1530  nfnt  1543  nf2  1555  drnf1  1618  drnf2  1619  nfexd  1641  dveeq2or  1694  nfsb2or  1715  nfdv  1754  nfsbxy  1815  nfsbxyt  1816  sbcomxyyz  1843  sbnf2  1854  dvelimALT  1883  dvelimfv  1884  nfsb4t  1887  dvelimor  1891  oprabidlem  5479  bj-nfalt  9219
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