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Mirrors > Home > ILE Home > Th. List > rabeqf | GIF version |
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
Ref | Expression |
---|---|
rabeqf.1 | ⊢ ℲxA |
rabeqf.2 | ⊢ ℲxB |
Ref | Expression |
---|---|
rabeqf | ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqf.1 | . . . 4 ⊢ ℲxA | |
2 | rabeqf.2 | . . . 4 ⊢ ℲxB | |
3 | 1, 2 | nfeq 2182 | . . 3 ⊢ Ⅎx A = B |
4 | eleq2 2098 | . . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B)) | |
5 | 4 | anbi1d 438 | . . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ))) |
6 | 3, 5 | abbid 2151 | . 2 ⊢ (A = B → {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ B ∧ φ)}) |
7 | df-rab 2309 | . 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
8 | df-rab 2309 | . 2 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)} | |
9 | 6, 7, 8 | 3eqtr4g 2094 | 1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 {cab 2023 Ⅎwnfc 2162 {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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 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-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 |
This theorem is referenced by: rabeq 2545 |
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