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Definition df-nf 1391
Description: Define the not-free predicate for wffs. This is read " x is not free in  ph". Not-free means that the value of  x cannot affect the value of  ph, e.g., any occurrence of  x in  ph 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 1702). An example of where this is used is stdpc5 1517. See nf2 1599 for an alternate 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,  x is effectively not free in the bare expression  x  =  x, even though  x would be considered free in the usual textbook definition, because the value of  x in the expression  x  =  x 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/ x ph  <->  A. x
( ph  ->  A. x ph ) )

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  setvar  x
31, 2wnf 1390 . 2  wff  F/ x ph
41, 2wal 1283 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1283 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 103 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1392  nfbii  1403  nfr  1452  nfd  1457  nfbidf  1473  nfnf1  1477  nford  1500  nfand  1501  nfnf  1510  nfalt  1511  19.21t  1515  nfimd  1518  19.9t  1574  nfnt  1587  nf2  1599  drnf1  1663  drnf2  1664  nfexd  1686  dveeq2or  1739  nfsb2or  1760  nfdv  1800  nfsbxy  1861  nfsbxyt  1862  sbcomxyyz  1889  sbnf2  1900  dvelimALT  1929  dvelimfv  1930  nfsb4t  1933  dvelimor  1937  oprabidlem  5587  bj-nfalt  10835
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