Description: Define proper substitution. For our notation, we use [ t / x ] ph to mean "the wff that results from the proper substitution of t for x in the wff ph ". That is, t properly replaces x . For example, [ t / x ] z e. x is the same as z e. t (when x and z are distinct), as shown in elsb2 .
Our notation was introduced in Haskell B. Curry'sFoundations 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, " ph ( t ) is the wff that results when t is properly substituted for x in ph ( x ) ". For example, if the original ph ( x ) is x = t , then ph ( t ) is t = t , from which we obtain that ph ( x ) is x = x . So what exactly does ph ( x ) mean? Curry's notation solves this problem.
A very similar notation, namely ( y | x ) ph , was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953).
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 , sbcom2 and sbid2v ).
Note that our definition is valid even when x and t are replaced with the same variable, as sbid shows. We achieve this by applying twice Tarski's definition sb6 which is valid for disjoint variables, and introducing a dummy variable y which isolates x from t , as in dfsb7 with respect to sb5 . We can also achieve this by having x free in the first conjunct and bound in the second, as the alternate definition dfsb1 shows. Another version that mixes free and bound variables is dfsb3 . When x and t are distinct, we can express proper substitution with the simpler expressions of sb5 and sb6 .
Note that the occurrences of a given variable in the definiens are either all bound ( x , y ) or all free ( t ). Also note that the definiens uses only primitive symbols.
This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. (Contributed by NM, 10-May-1993) Revised from the original definition dfsb1 . (Revised by BJ, 22-Dec-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-sb | ⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vt | ⊢ 𝑡 | |
| 1 | vx | ⊢ 𝑥 | |
| 2 | wph | ⊢ 𝜑 | |
| 3 | 2 1 0 | wsb | ⊢ [ 𝑡 / 𝑥 ] 𝜑 |
| 4 | vy | ⊢ 𝑦 | |
| 5 | 4 | cv | ⊢ 𝑦 |
| 6 | 0 | cv | ⊢ 𝑡 |
| 7 | 5 6 | wceq | ⊢ 𝑦 = 𝑡 |
| 8 | 1 | cv | ⊢ 𝑥 |
| 9 | 8 5 | wceq | ⊢ 𝑥 = 𝑦 |
| 10 | 9 2 | wi | ⊢ ( 𝑥 = 𝑦 → 𝜑 ) |
| 11 | 10 1 | wal | ⊢ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) |
| 12 | 7 11 | wi | ⊢ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 13 | 12 4 | wal | ⊢ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 14 | 3 13 | wb | ⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |