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Definition df-nf 1326
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 1638). An example of where this is used is stdpc5 1454. See nf2 1536 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 1325 . 2  F/
41, 2wal 1224 . . . 4
51, 4wi 4 . . 3
65, 2wal 1224 . 2
73, 6wb 98 1  F/
Colors of variables: wff set class
This definition is referenced by:  nfi  1327  nfbii  1338  nfr  1388  nfd  1393  nfbidf  1410  nfnf1  1414  nford  1437  nfand  1438  nfnf  1447  nfalt  1448  19.21t  1452  nfimd  1455  19.9t  1511  nfnt  1524  nf2  1536  drnf1  1599  drnf2  1600  nfexd  1622  dveeq2or  1675  nfsb2or  1696  nfdv  1735  nfsbxy  1796  nfsbxyt  1797  sbcomxyyz  1824  sbnf2  1835  dvelimALT  1864  dvelimfv  1865  nfsb4t  1868  dvelimor  1872  oprabidlem  5456  bj-nfalt  7203
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