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Definition df-nf 1330
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 1642). An example of where this is used is stdpc5 1458. See nf2 1540 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 1329 . 2 wff xφ
41, 2wal 1226 . . . 4 wff xφ
51, 4wi 4 . . 3 wff (φxφ)
65, 2wal 1226 . 2 wff x(φxφ)
73, 6wb 98 1 wff (Ⅎxφx(φxφ))
Colors of variables: wff set class
This definition is referenced by:  nfi  1331  nfbii  1342  nfr  1392  nfd  1397  nfbidf  1414  nfnf1  1418  nford  1441  nfand  1442  nfnf  1451  nfalt  1452  19.21t  1456  nfimd  1459  19.9t  1515  nfnt  1528  nf2  1540  drnf1  1603  drnf2  1604  nfexd  1626  dveeq2or  1679  nfsb2or  1700  nfdv  1739  nfsbxy  1800  nfsbxyt  1801  sbcomxyyz  1828  sbnf2  1839  dvelimALT  1868  dvelimfv  1869  nfsb4t  1872  dvelimor  1876  oprabidlem  5460  bj-nfalt  7011
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