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Definition df-nf 1405
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 1718). An example of where this is used is stdpc5 1531. See nf2 1614 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 1404 . 2  wff  F/ x ph
41, 2wal 1297 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1297 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 104 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1406  nfbii  1417  nfr  1466  nfd  1471  nfbidf  1487  nfnf1  1491  nford  1514  nfand  1515  nfnf  1524  nfalt  1525  19.21t  1529  nfimd  1532  19.9t  1589  nfnt  1602  nf2  1614  drnf1  1679  drnf2  1680  nfexd  1702  dveeq2or  1755  nfsb2or  1776  nfdv  1816  nfsbxy  1878  nfsbxyt  1879  sbcomxyyz  1906  sbnf2  1917  dvelimALT  1946  dvelimfv  1947  nfsb4t  1950  dvelimor  1954  oprabidlem  5734  bj-nfalt  12553
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