Definition df-sb 1209

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 1222.

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 1266, sbcom2 1373 and sbid2v 1382).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1221 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 1379 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 1263. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1306 and sb6 1305.

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 991	. . 3 class y
5	1, 2, 4	wsbc 1207	. 2 wff [y / x]φ
6	2	cv 991	. . . . 5 class x
7	6, 4	wceq 992	. . . 4 wff x = y
8	7, 1	wi 3	. . 3 wff (x = y → φ)
9	7, 1	wa 221	. . . 4 wff (x = y ⋀ φ)
10	9, 2	wex 1016	. . 3 wff ∃x(x = y ⋀ φ)
11	8, 10	wa 221	. 2 wff ((x = y → φ) ⋀ ∃x(x = y ⋀ φ))
12	5, 11	wb 144	1 wff ([y / x]φ ↔ ((x = y → φ) ⋀ ∃x(x = y ⋀ φ)))

This definition is referenced by:  sbimi 1210  drsb1 1212  sb1 1213  sb2 1214  sbequ1 1215  sbequ2 1216  sbn 1268  sb6 1305

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