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Mirrors > Home > ILE Home > Th. List > rabn0m | GIF version |
Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.) |
Ref | Expression |
---|---|
rabn0m | ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2355 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | rabid 2530 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | 2 | exbii 1537 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
4 | nfv 1462 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} | |
5 | df-rab 2358 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 5 | eleq2i 2146 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
7 | nfsab1 2072 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
8 | 6, 7 | nfxfr 1404 | . . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} |
9 | eleq1 2142 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) | |
10 | 4, 8, 9 | cbvex 1680 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
11 | 1, 3, 10 | 3bitr2ri 207 | 1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1422 ∈ wcel 1434 {cab 2068 ∃wrex 2350 {crab 2353 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-11 1438 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-rex 2355 df-rab 2358 |
This theorem is referenced by: exss 3984 |
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