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Definition df-nf 1555
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 2119). An example of where this is used is stdpc5 1817. See nf2 1890 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 1691), 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 2563 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 1554 . 2  wff  F/ x ph
41, 2wal 1550 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1550 . 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  1561  nfbii  1579  nfdv  1650  nfr  1778  nfd  1783  nfbidf  1791  19.9t  1794  nfnf1  1809  nfnd  1810  nfndOLD  1811  nfimd  1828  nfimdOLD  1829  nfnf  1868  nfnfOLD  1869  nf2  1890  drnf1  2062  drnf2OLD  2064  sbnf2  2186  axie2  2414  xfree  23952  hbexg  28717  drnf1NEW7  29569  drnf2wAUX7  29572  drnf2w2AUX7  29575  drnf2w3AUX7  29578  sbnf2NEW7  29682  nfnfOLD7  29763  drnf2OLD7  29790
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