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Mirrors > Home > ILE Home > Th. List > abeq2 | GIF version |
Description: Equality of a class
variable and a class abstraction (also called a
class builder). Theorem 5.1 of [Quine] p.
34. This theorem shows the
relationship between expressions with class abstractions and expressions
with class variables. Note that abbi 2226 and its relatives are among
those useful for converting theorems with class variables to equivalent
theorems with wff variables, by first substituting a class abstraction
for each class variable.
Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
abeq2 | ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1487 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
2 | hbab1 2102 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | cleqh 2212 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) |
4 | abid 2101 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | 4 | bibi2i 226 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) |
6 | 5 | albii 1427 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
7 | 3, 6 | bitri 183 | 1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1310 = wceq 1312 ∈ wcel 1461 {cab 2099 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 |
This theorem is referenced by: abeq1 2222 abbi2i 2227 abbi2dv 2231 clabel 2238 sbabel 2279 rabid2 2579 ru 2875 sbcabel 2956 dfss2 3050 vpwex 4061 dmopab3 4710 |
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