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Definition df-nf 1535
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 1979). An example of where this is used is stdpc5 1805. See nf2 1810 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 1663), 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 2421 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

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 1534 . 2  wff  F/ x ph
41, 2wal 1530 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1530 . 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  1541  nfbii  1559  nfdv  1629  nfr  1753  nfd  1758  nfbidf  1766  nfnf1  1769  nfnd  1772  nfimd  1773  nfnf  1780  nf2  1810  drnf1  1922  drnf2  1923  sbnf2  2060  xfree  23040  hbexg  28621  drnf1NEW7  29472  drnf2wAUX7  29475  drnf2w2AUX7  29478  drnf2w3AUX7  29481  sbnf2NEW7  29580  nfnfOLD7  29643  drnf2OLD7  29670
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