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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 2148 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 1416 | . . 3 ⊢ (y ∈ A → ∀x y ∈ A) | |
2 | hbab1 2026 | . . 3 ⊢ (y ∈ {x ∣ φ} → ∀x y ∈ {x ∣ φ}) | |
3 | 1, 2 | cleqh 2134 | . 2 ⊢ (A = {x ∣ φ} ↔ ∀x(x ∈ A ↔ x ∈ {x ∣ φ})) |
4 | abid 2025 | . . . 4 ⊢ (x ∈ {x ∣ φ} ↔ φ) | |
5 | 4 | bibi2i 216 | . . 3 ⊢ ((x ∈ A ↔ x ∈ {x ∣ φ}) ↔ (x ∈ A ↔ φ)) |
6 | 5 | albii 1356 | . 2 ⊢ (∀x(x ∈ A ↔ x ∈ {x ∣ φ}) ↔ ∀x(x ∈ A ↔ φ)) |
7 | 3, 6 | bitri 173 | 1 ⊢ (A = {x ∣ φ} ↔ ∀x(x ∈ A ↔ φ)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 {cab 2023 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 |
This theorem is referenced by: abeq1 2144 abbi2i 2149 abbi2dv 2153 clabel 2160 sbabel 2200 rabid2 2480 ru 2757 sbcabel 2833 dfss2 2928 pwex 3923 dmopab3 4491 |
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