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Definition df-nf 1532
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 1966). An example of where this is used is stdpc5 1793. See nf2 1798 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 (see nfequid 1645), 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 2408 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 1531 . 2  wff  F/ x ph
41, 2wal 1527 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1527 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 176 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1538  nfbii  1556  nfdv  1620  nfr  1741  nfd  1746  nfbidf  1754  nfnf1  1757  nfnd  1760  nfimd  1761  nfnf  1768  nf2  1798  drnf1  1909  drnf2  1910  sbnf2  2047  xfree  23024  hbexg  28322
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