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Definition df-sb 1677
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 2199. We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2099.

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 2121, sbcom2 2210 and sbid2v 2227).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1951 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 2225 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 2119. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2194 and sb6 2192.

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 1676 . 2
52, 3weq 1671 . . . 4
65, 1wi 4 . . 3
75, 1wa 360 . . . 4
87, 2wex 1565 . . 3
96, 8wa 360 . 2
104, 9wb 178 1
Colors of variables: wff set class
This definition is referenced by:  sbequ2  1678  sbequ2OLD  1679  sb1  1680  sbequ8  1682  sbimi  1683  sbequ1  1947  sb2  2098  drsb1  2122  sbn  2140  sbnOLD  2141  sb6OLD  2193  subsym1  27516  sbcom3OLD  28123  drsb1NEW11  31058  sb2NEW11  31089  sbnNEW11  31114  sb6NEW11  31149
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