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Definition df-sb 1661
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 ." We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2095.

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 2116, sbcom2 2197 and sbid2v 2207).

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 2205 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 2114. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2183 and sb6 2182.

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 1660 . 2
52, 3weq 1655 . . . 4
65, 1wi 4 . . 3
75, 1wa 360 . . . 4
87, 2wex 1551 . . 3
96, 8wa 360 . 2
104, 9wb 178 1
Colors of variables: wff set class
This definition is referenced by:  sbequ2  1662  sbequ2OLD  1663  sb1  1664  sbequ8  1666  sbimi  1667  sbequ1  1947  sb2  2094  drsb1  2117  sbn  2135  sbnOLD  2136  sbrim  2141  sb6  2182  subsym1  26281  sbcom3  26864  drsb1NEW7  29747  sb2NEW7  29778  sbnNEW7  29803  sb6NEW7  29838
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