![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rabid2 | GIF version |
Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rabid2 | ⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2 2143 | . . 3 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ))) | |
2 | pm4.71 369 | . . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ∧ φ))) | |
3 | 2 | albii 1356 | . . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ))) |
4 | 1, 3 | bitr4i 176 | . 2 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A → φ)) |
5 | df-rab 2309 | . . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
6 | 5 | eqeq2i 2047 | . 2 ⊢ (A = {x ∈ A ∣ φ} ↔ A = {x ∣ (x ∈ A ∧ φ)}) |
7 | df-ral 2305 | . 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
8 | 4, 6, 7 | 3bitr4i 201 | 1 ⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 {cab 2023 ∀wral 2300 {crab 2304 |
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 df-ral 2305 df-rab 2309 |
This theorem is referenced by: rabxmdc 3243 rabrsndc 3429 class2seteq 3907 dmmptg 4761 fneqeql 5218 fmpt 5262 acexmidlemph 5448 ioomax 8587 iccmax 8588 |
Copyright terms: Public domain | W3C validator |