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Mirrors > Home > ILE Home > Th. List > abn0m | GIF version |
Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.) |
Ref | Expression |
---|---|
abn0m | ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1521 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑} | |
2 | nfsab1 2160 | . . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
3 | eleq1w 2231 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
4 | 1, 2, 3 | cbvex 1749 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) |
5 | abid 2158 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | 5 | exbii 1598 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
7 | 4, 6 | bitr3i 185 | 1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1485 ∈ wcel 2141 {cab 2156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-clel 2166 |
This theorem is referenced by: mapprc 6630 |
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