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Definition df-nf 1533
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 1972). An example of where this is used is stdpc5 1795. See nf2 1800 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 2103), 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 2410 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 1532 . 2  wff  F/ x ph
41, 2wal 1528 . . . 4  wff  A. x ph
51, 4wi 6 . . 3  wff  ( ph  ->  A. x ph )
65, 2wal 1528 . 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  1539  nfbii  1557  nfdv  1621  nfr  1743  nfd  1748  nfbidf  1756  nfnf1  1759  nfnd  1762  nfimd  1763  nfnf  1770  nf2  1800  drnf1  1912  drnf2  1913  sbnf2  2048  xfree  23017  hbexg  27594
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