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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 2122). An example of where this is used is stdpc5 1819. See nf2 1892 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 1693), 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 2568 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 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  1651  nfr  1780  nfd  1785  nfbidf  1793  19.9t  1796  nfnf1  1811  nfnd  1812  nfndOLD  1813  nfimd  1830  nfimdOLD  1831  nfnf  1870  nfnfOLD  1871  nf2  1892  drnf1  2065  drnf2OLD  2067  sbnf2  2191  axie2  2419  xfree  23985  hbexg  28815  drnf1NEW7  29669  drnf2wAUX7  29672  drnf2w2AUX7  29675  drnf2w3AUX7  29678  sbnf2NEW7  29782  nfnfOLD7  29863  drnf2OLD7  29890
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