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Definition df-nf 1540
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 1898). An example of where this is used is stdpc5 1773.

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 (see nfequid 1688), 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).

This predicate only applies to wffs. See df-nfc 2374 for a not-free predicate for class variables. (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  set  x
31, 2wnf 1539 . 2  wff  F/ x ph
41, 2wal 1532 . . . 4  wff  A. x ph
51, 4wi 6 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1532 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 178 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1556  nfbii  1561  nfr  1702  nfd  1707  nfbidf  1715  nfnf1  1720  nfnd  1726  nfimd  1727  nfnf  1734  nf2  1778  drnf1  1861  drnf2  1862  nfdv  2015  sbnf2  2068  xfree  22854  hbexg  27015
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