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Definition df-nf 1551
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 2079). An example of where this is used is stdpc5 1812. See nf2 1885 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 1686), 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 2533 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 1550 . 2  wff  F/ x ph
41, 2wal 1546 . . . 4  wff  A. x ph
51, 4wi 4 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1546 . 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  1557  nfbii  1575  nfdv  1646  nfr  1773  nfd  1778  nfbidf  1786  19.9t  1789  nfnf1  1804  nfnd  1805  nfndOLD  1806  nfimd  1823  nfimdOLD  1824  nfnf  1863  nfnfOLD  1864  nf2  1885  drnf1  2029  drnf2  2030  sbnf2  2161  axie2  2383  xfree  23904  hbexg  28358  drnf1NEW7  29205  drnf2wAUX7  29208  drnf2w2AUX7  29211  drnf2w3AUX7  29214  sbnf2NEW7  29313  nfnfOLD7  29377  drnf2OLD7  29404
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