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Mirrors > Home > MPE Home > Th. List > df-sb | Structured version Visualization version GIF version |
Description: Define proper
substitution. For our notation, we use [𝑡 / 𝑥]𝜑
to mean "the wff that results from the proper substitution of 𝑡 for
𝑥 in the wff 𝜑". That is, 𝑡
properly replaces 𝑥.
For example, [𝑡 / 𝑥]𝑧 ∈ 𝑥 is the same as 𝑧 ∈ 𝑡 (when 𝑥
and 𝑧 are distinct), as shown in elsb2 2115.
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(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem. A very similar notation, namely (𝑦 ∣ 𝑥)𝜑, 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 2078, sbcom2 2153 and sbid2v 2500). Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2239 shows. We achieve this by applying twice Tarski's definition sb6 2080 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2267 with respect to sb5 2259. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2472 shows. Another version that mixes free and bound variables is dfsb3 2485. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2259 and sb6 2080. 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 2472. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
df-sb | ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vt | . . 3 setvar 𝑡 | |
4 | 1, 2, 3 | wsb 2059 | . 2 wff [𝑡 / 𝑥]𝜑 |
5 | vy | . . . . 5 setvar 𝑦 | |
6 | 5, 3 | weq 1958 | . . . 4 wff 𝑦 = 𝑡 |
7 | 2, 5 | weq 1958 | . . . . . 6 wff 𝑥 = 𝑦 |
8 | 7, 1 | wi 4 | . . . . 5 wff (𝑥 = 𝑦 → 𝜑) |
9 | 8, 2 | wal 1531 | . . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑) |
10 | 6, 9 | wi 4 | . . 3 wff (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
11 | 10, 5 | wal 1531 | . 2 wff ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
12 | 4, 11 | wb 205 | 1 wff ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
This definition is referenced by: sbt 2061 stdpc4 2063 sbi1 2066 spsbe 2077 sbequ 2078 sb6 2080 sbal 2151 hbsbw 2161 sbequ1 2232 sbequ2 2233 dfsb7 2267 sbn 2268 sbrim 2292 nfsbvOLD 2316 cbvsbvf 2352 sb4b 2466 bj-ssbeq 36031 bj-ssbid2ALT 36041 bj-ssbid1ALT 36043 |
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