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Definition df-sb 1669
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 2091.

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 2113, sbcom2 2202 and sbid2v 2219).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1943 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 2217 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 2111. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2186 and sb6 2184.

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 1668 . 2
52, 3weq 1663 . . . 4
65, 1wi 4 . . 3
75, 1wa 360 . . . 4
87, 2wex 1557 . . 3
96, 8wa 360 . 2
104, 9wb 178 1
Colors of variables: wff set class
This definition is referenced by:  sbequ2  1670  sbequ2OLD  1671  sb1  1672  sbequ8  1674  sbimi  1675  sbequ1  1939  sb2  2090  drsb1  2114  sbn  2132  sbnOLD  2133  sb6OLD  2185  subsym1  27007  sbcom3OLD  27614  drsb1NEW11  30781  sb2NEW11  30812  sbnNEW11  30837  sb6NEW11  30872
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