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Definition df-nf 1569
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 2127). An example of where this is used is stdpc5 1822. See nf2 1892 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 1707), 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 2614 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

Ref Expression

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2wnf 1568 . 2
41, 2wal 1564 . . . 4
51, 4wi 4 . . 3
65, 2wal 1564 . 2
73, 6wb 178 1
Colors of variables: wff set class
This definition is referenced by:  nfi  1575  nfbii  1593  nfdv  1667  nfr  1784  nfd  1789  nfbidf  1797  19.9t  1800  nfnf1  1814  nfnd  1815  nfimd  1833  nfimdOLD  1834  nfnf  1870  nfnfOLD  1871  nf2  1892  drnf1  2069  drnf2OLD  2071  sbnf2OLD  2204  axie2  2465  xfree  24970  hbexg  30111  drnf1NEW11  31047  drnf2wAUX11  31050  drnf2w2AUX11  31053  drnf2w3AUX11  31056  sbnf2NEW11  31159  nfnfOLD11  31240  drnf2OLD11  31267
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