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

Definition df-sb 1831
Description: Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑦 for 𝑥 in the wff 𝜑." That is, 𝑦 properly replaces 𝑥. For example, [𝑥 / 𝑦]𝑧𝑦 is the same as 𝑧𝑥, as shown in elsb4 2327. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2245.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)." For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2268, sbcom2 2337 and sbid2v 2349).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2133 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2347 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2266. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2322 and sb6 2321.

There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 10-May-1993.)

Assertion
Ref Expression
df-sb ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
41, 2, 3wsb 1830 . 2 wff [𝑦 / 𝑥]𝜑
52, 3weq 1824 . . . 4 wff 𝑥 = 𝑦
65, 1wi 4 . . 3 wff (𝑥 = 𝑦𝜑)
75, 1wa 382 . . . 4 wff (𝑥 = 𝑦𝜑)
87, 2wex 1694 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
96, 8wa 382 . 2 wff ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))
104, 9wb 194 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
This definition is referenced by:  sbequ2  1832  sb1  1833  sbequ8  1835  sbimi  1836  sbequ1  2129  sb2  2244  drsb1  2269  sbn  2283  subsym1  31431  bj-sb2v  31779  bj-dfsb2  31855  frege55b  37093
  Copyright terms: Public domain W3C validator