Definition df-sb 1168

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 1181.

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 remarkable little formula that is exactly equivalent and gives us a single direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1224, sbcom2 1329 and sbid2v 1338).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1180 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 1335 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 1221. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1263 and sb6 1262.

There are no restrictions on any of the variables, including what variables may occur in wff φ.

Assertion

Ref	Expression
df-sb	⊢ ([y / x]φ ↔ ((x = y → φ) ⋀ ∃x(x = y ⋀ φ)))

Detailed syntax breakdown of Definition df-sb

Step	Hyp	Ref	Expression
1		wph	. . 3 wff φ
2		vx	. . 3 set x
3		vy	. . . 4 set y
4	3	cv 952	. . 3 class y
5	1, 2, 4	wsbc 1166	. 2 wff [y / x]φ
6	2	cv 952	. . . . 5 class x
7	6, 4	wceq 953	. . . 4 wff x = y
8	7, 1	wi 3	. . 3 wff (x = y → φ)
9	7, 1	wa 223	. . . 4 wff (x = y ⋀ φ)
10	9, 2	wex 977	. . 3 wff ∃x(x = y ⋀ φ)
11	8, 10	wa 223	. 2 wff ((x = y → φ) ⋀ ∃x(x = y ⋀ φ))
12	5, 11	wb 146	1 wff ([y / x]φ ↔ ((x = y → φ) ⋀ ∃x(x = y ⋀ φ)))

This definition is referenced by:  sbimi 1169  drsb1 1171  sb1 1172  sb2 1173  sbequ1 1174  sbequ2 1175  sbn 1226  sb6 1262

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