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Mirrors > Home > ILE Home > Th. List > dfsb  Structured version GIF version 
Description: Define proper
substitution. Remark 9.1 in [Megill] p. 447 (p.
15 of the
preprint). For our notation, we use [y / x]φ to mean "the wff
that results when y is
properly substituted for x in
the wff
φ." We can
also use [y / x]φ
in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1598.
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, "φ(y) is the wff that results when y is properly substituted for x in φ(x)." For example, if the original φ(x) is x = y, then φ(y) is y = y, from which we obtain that φ(x) is x = x. So what exactly does φ(x) 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 1660, sbcom2 1798 and sbid2v 1807). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1597 shows. We achieve this by having x 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 1802 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 1805. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1705 and sb6 1704. In classical logic, another possible definition is (x = y ∧ φ) ∨ ∀x(x = y → φ) 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 φ. (Contributed by NM, 5Aug1993.) 
Ref  Expression 

dfsb  ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) 
Step  Hyp  Ref  Expression 

1  wph  . . 3 wff φ  
2  vx  . . 3 set x  
3  vy  . . 3 set y  
4  1, 2, 3  wsb 1585  . 2 wff [y / x]φ 
5  2, 3  weq 1341  . . . 4 wff x = y 
6  5, 1  wi 4  . . 3 wff (x = y → φ) 
7  5, 1  wa 95  . . . 4 wff (x = y ∧ φ) 
8  7, 2  wex 1328  . . 3 wff ∃x(x = y ∧ φ) 
9  6, 8  wa 95  . 2 wff ((x = y → φ) ∧ ∃x(x = y ∧ φ)) 
10  4, 9  wb 96  1 wff ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) 
Colors of variables: wff set class 
This definition is referenced by: sbimi 1587 sb1 1589 sb2 1590 sbequ1 1591 sbequ2 1592 drsb1 1619 spsbim 1663 sbequ8 1666 sbidm 1670 sb6 1704 hbsbv 1752 
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