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Definition df-nf 1323
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 1633). An example of where this is used is stdpc5 1449. See nf2 1531 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 , 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). (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf  F/

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3
2 vx . . 3  setvar
31, 2wnf 1322 . 2  F/
41, 2wal 1221 . . . 4
51, 4wi 4 . . 3
65, 2wal 1221 . 2
73, 6wb 98 1  F/
Colors of variables: wff set class
This definition is referenced by:  nfi  1324  nfbii  1335  nfr  1384  nfd  1389  nfbidf  1405  nfnf1  1409  nford  1432  nfand  1433  nfnf  1442  nfalt  1443  19.21t  1447  nfimd  1450  19.9t  1506  nfnt  1519  nf2  1531  drnf1  1594  drnf2  1595  nfexd  1617  dveeq2or  1670  nfsb2or  1691  nfdv  1730  nfsbxy  1791  nfsbxyt  1792  sbcomxyyz  1819  sbnf2  1830  dvelimALT  1859  dvelimfv  1860  nfsb4t  1863  dvelimor  1867  oprabidlem  5448  bj-nfalt  8169
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