MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-nf Structured version   Visualization version   GIF version

Definition df-nf 1778
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 2254). An example of where this is used is stdpc5 2193. See nf5 2270 for an alternate definition which involves 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 considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered free in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 2008.

This definition of "not free" tightly ties to the quantifier 𝑥. At this state (no axioms restricting quantifiers yet) "nonfree" appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization.

The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1972.

This predicate only applies to wffs. See df-nfc 2877 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.)

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

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
31, 2wnf 1777 . 2 wff 𝑥𝜑
41, 2wex 1773 . . 3 wff 𝑥𝜑
51, 2wal 1531 . . 3 wff 𝑥𝜑
64, 5wi 4 . 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
73, 6wb 205 1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nf2  1779  nfi  1782  nfri  1783  nfd  1784  nfrd  1785  nfbiit  1845  nfnbi  1849  nfbidv  1917  nfnf1  2143  nfbidf  2209  nf5  2270  nf6  2271  nfnf  2311  sbnf2  2346  drnf1v  2361  nfcriOLD  2885  nfcriOLDOLD  2886  bj-nfimt  36015  bj-nnfnfTEMP  36116  bj-nfnnfTEMP  36136
  Copyright terms: Public domain W3C validator