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Definition df-nf 1545
Description: Define the not-free predicate for wffs. This is read " is not free in ". Not-free means that the value of cannot affect the value of , e.g., any occurrence of in 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 2026). An example of where this is used is stdpc5 1798. See nf2 1866 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, is effectively not free in the bare expression (see nfequid 1678), even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result).

This predicate only applies to wffs. See df-nfc 2478 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

Ref Expression
df-nf  F/

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2wnf 1544 . 2  F/
41, 2wal 1540 . . . 4
51, 4wi 4 . . 3
65, 2wal 1540 . 2
73, 6wb 176 1  F/
Colors of variables: wff setvar class
This definition is referenced by:  nfi  1551  nfbii  1569  nfdv  1639  nfr  1761  nfd  1766  nfbidf  1774  19.9t  1779  nfnf1  1790  nfnd  1791  nfndOLD  1792  nfimd  1808  nfimdOLD  1809  nfnf  1845  nfnfOLD  1846  nf2  1866  drnf1  1969  drnf2  1970  sbnf2  2108  axie2  2329
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