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Definition df-sb 1703
Description: Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." That is, properly replaces . For example, is the same as , as shown in elsb4 2149. We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2054.

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, " (y) is the wff that results when is properly substituted for in (x)." For example, if the original (x) is , then (y) is , from which we obtain that (x) is . So what exactly does (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 2077, sbcom2 2160 and sbid2v 2176).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1952 shows. We achieve this by having 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 2174 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 2075. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2144 and sb6 2142.

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

Ref Expression

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 vy . . 3
41, 2, 3wsb 1702 . 2
52, 3weq 1696 . . . 4
65, 1wi 4 . . 3
75, 1wa 369 . . . 4
87, 2wex 1587 . . 3
96, 8wa 369 . 2
104, 9wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  sbequ2  1704  sb1  1705  sbequ8  1707  sbimi  1708  sbequ1  1947  sb2  2053  drsb1  2078  sbn  2092  sbnOLD  2093  sb6OLD  2143  subsym1  28729  bj-sb2v  33113  frege55b  36578  frege58b  36581
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