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Definition df-sb 1740
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 2179. We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2094.

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, " ( ) is the wff that results when is properly substituted for in (x)." For example, if the original (x) is , then ( ) 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 2117, sbcom2 2189 and sbid2v 2201).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1996 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 2199 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 2115. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2174 and sb6 2173.

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 1739 . 2
52, 3weq 1733 . . . 4
65, 1wi 4 . . 3
75, 1wa 369 . . . 4
87, 2wex 1612 . . 3
96, 8wa 369 . 2
104, 9wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  sbequ2  1741  sb1  1742  sbequ8  1744  sbimi  1745  sbequ1  1991  sb2  2093  drsb1  2118  sbn  2132  subsym1  29892  bj-sb2v  34333  frege55b  37924
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