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Definition df-nf 1561
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 2134). An example of where this is used is stdpc5 1827. See nf2 1900 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 1701), 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 2604 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 set 𝑥
31, 2wnf 1560 . 2 wff 𝑥𝜑
41, 2wal 1556 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1556 . 2 wff 𝑥(𝜑 → ∀𝑥𝜑)
73, 6wb 178 1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
This definition is referenced by:  nfi  1567  nfbii  1585  nfdv  1659  nfr  1787  nfd  1792  nfbidf  1800  19.9t  1803  nfnf1  1818  nfnd  1819  nfndOLD  1820  nfimd  1838  nfimdOLD  1839  nfnf  1878  nfnfOLD  1879  nf2  1900  drnf1  2077  drnf2OLD  2079  sbnf2OLD  2211  axie2  2455  xfree  24079  hbexg  29609  drnf1NEW7  30545  drnf2wAUX7  30548  drnf2w2AUX7  30551  drnf2w3AUX7  30554  sbnf2NEW7  30658  nfnfOLD7  30739  drnf2OLD7  30766
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