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Definition df-sb 1628
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 1640.

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 1703, sbcom2 1845 and sbid2v 1854).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1639 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 1849 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 1852.

When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1749 and sb6 1748.

In classical logic, another possible definition is (x = y φ) x(x = yφ) 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, 5-Aug-1993.)

Ref Expression
df-sb ([y / x]φ ↔ ((x = yφ) x(x = y φ)))

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 setvar x
3 vy . . 3 setvar y
41, 2, 3wsb 1627 . 2 wff [y / x]φ
52, 3weq 1373 . . . 4 wff x = y
65, 1wi 4 . . 3 wff (x = yφ)
75, 1wa 97 . . . 4 wff (x = y φ)
87, 2wex 1362 . . 3 wff x(x = y φ)
96, 8wa 97 . 2 wff ((x = yφ) x(x = y φ))
104, 9wb 98 1 wff ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
Colors of variables: wff set class
This definition is referenced by:  sbimi  1629  sb1  1631  sb2  1632  sbequ1  1633  sbequ2  1634  drsb1  1662  spsbim  1706  sbequ8  1709  sbidm  1713  sb6  1748  hbsbv  1799
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