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Definition df-nf 1617
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 2121). An example of where this is used is stdpc5 1908. See nf2 1960 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 1792), 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 2607 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2wnf 1616 . 2
41, 2wal 1393 . . . 4
51, 4wi 4 . . 3
65, 2wal 1393 . 2
73, 6wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  nfi  1623  nfbii  1644  nfdv  1727  nfr  1873  nfd  1878  nfbidf  1887  19.9t  1890  nfnf1  1899  nfnt  1900  nfimd  1917  nfnf  1949  nf2  1960  drnf1  2071  axie2  2430  xfree  27363  wl-sbnf1  30003  hbexg  33329  bj-nfdt0  34248  bj-nfalt  34265  bj-nfext  34266  bj-nfs1t  34274  bj-drnf1v  34328  bj-sbnf  34412
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