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Definition df-nf 1366
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 1676). An example of where this is used is stdpc5 1492. See nf2 1574 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,  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 1365 . 2  wff  F/ x ph
41, 2wal 1257 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1257 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 102 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1367  nfbii  1378  nfr  1427  nfd  1432  nfbidf  1448  nfnf1  1452  nford  1475  nfand  1476  nfnf  1485  nfalt  1486  19.21t  1490  nfimd  1493  19.9t  1549  nfnt  1562  nf2  1574  drnf1  1637  drnf2  1638  nfexd  1660  dveeq2or  1713  nfsb2or  1734  nfdv  1773  nfsbxy  1834  nfsbxyt  1835  sbcomxyyz  1862  sbnf2  1873  dvelimALT  1902  dvelimfv  1903  nfsb4t  1906  dvelimor  1910  oprabidlem  5563  bj-nfalt  10270
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