ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-nf Structured version   GIF version

Definition df-nf 1330
Description: Define the not-free predicate for wffs. This is read "x is not free in φ". Not-free means that the value of x cannot affect the value of φ, e.g., any occurrence of x in φ 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 1643). An example of where this is used is stdpc5 1460. See nf2 1540 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, 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). (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf (Ⅎxφx(φxφ))

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 setvar x
31, 2wnf 1329 . 2 wff xφ
41, 2wal 1315 . . . 4 wff xφ
51, 4wi 4 . . 3 wff (φxφ)
65, 2wal 1315 . 2 wff x(φxφ)
73, 6wb 98 1 wff (Ⅎxφx(φxφ))
Colors of variables: wff set class
This definition is referenced by:  nfi  1331  nfbii  1342  nfr  1394  nfd  1399  nfbidf  1416  nfnf1  1420  nfand  1444  nfnf  1453  nfalt  1454  19.21t  1458  nfimd  1461  19.9t  1517  nfnd  1530  nf2  1540  drnf1  1603  drnf2  1604  nfexd  1627  dveeq2or  1680  nfsb2or  1701  nfdv  1741  nfsbxy  1800  nfsbxyt  1801  sbcomxyyz  1828  sbnf2  1839  dvelimALT  1868  dvelimfv  1869  nfsb4t  1872  dvelimor  1876  oprabidlem  5439
  Copyright terms: Public domain W3C validator