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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 when
is properly substituted 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 1631.
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 1694, sbcom2 1836 and sbid2v 1845). Note that our definition is valid even when and are replaced with the same variable, as sbid 1630 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 1840 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1843. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1740 and sb6 1739. In classical logic, another possible definition is but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5Aug1993.) 
Ref  Expression 

dfsb 
Step  Hyp  Ref  Expression 

1  wph  . . 3  
2  vx  . . 3  
3  vy  . . 3  
4  1, 2, 3  wsb 1618  . 2 
5  2, 3  weq 1365  . . . 4 
6  5, 1  wi 4  . . 3 
7  5, 1  wa 97  . . . 4 
8  7, 2  wex 1354  . . 3 
9  6, 8  wa 97  . 2 
10  4, 9  wb 98  1 
Colors of variables: wff set class 
This definition is referenced by: sbimi 1620 sb1 1622 sb2 1623 sbequ1 1624 sbequ2 1625 drsb1 1653 spsbim 1697 sbequ8 1700 sbidm 1704 sb6 1739 hbsbv 1790 
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