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Definition df-nf 1554
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 2105). An example of where this is used is stdpc5 1816. See nf2 1889 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 1690), 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 2560 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 1553 . 2  wff  F/ x ph
41, 2wal 1549 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1549 . 2  wff  A. x
( ph  ->  A. x ph )
73, 6wb 177 1  wff  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
Colors of variables: wff set class
This definition is referenced by:  nfi  1560  nfbii  1578  nfdv  1649  nfr  1777  nfd  1782  nfbidf  1790  19.9t  1793  nfnf1  1808  nfnd  1809  nfndOLD  1810  nfimd  1827  nfimdOLD  1828  nfnf  1867  nfnfOLD  1868  nf2  1889  drnf1  2057  drnf2OLD  2059  sbnf2  2183  axie2  2411  xfree  23939  hbexg  28580  drnf1NEW7  29432  drnf2wAUX7  29435  drnf2w2AUX7  29438  drnf2w3AUX7  29441  sbnf2NEW7  29545  nfnfOLD7  29626  drnf2OLD7  29653
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