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Definition df-sb 1694
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 2132. We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2035.

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 ( )." For example, if the original ( ) is , then ( ) is , from which we obtain that ( ) is . So what exactly does ( ) 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 2056, sbcom2 2143 and sbid2v 2159).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1931 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 2157 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 2054. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2127 and sb6 2125.

There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
df-sb

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 vy . . 3
41, 2, 3wsb 1693 . 2
52, 3weq 1688 . . . 4
65, 1wi 4 . . 3
75, 1wa 362 . . . 4
87, 2wex 1581 . . 3
96, 8wa 362 . 2
104, 9wb 178 1
Colors of variables: wff setvar class
This definition is referenced by:  sbequ2  1695  sb1  1696  sbequ8  1698  sbimi  1699  sbequ1  1926  sb2  2034  drsb1  2057  sbn  2072  sbnOLD  2073  sb6OLD  2126  subsym1  27976  sbcom3OLD  28651  bj-sb2v  31773
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