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Mirrors > Home > ILE Home > Th. List > zfauscl | GIF version |
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4054, we invoke the Axiom of Extensionality (indirectly via vtocl 2743), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
zfauscl.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
zfauscl | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfauscl.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eleq2 2204 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴)) | |
3 | 2 | anbi1d 461 | . . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
4 | 3 | bibi2d 231 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
5 | 4 | albidv 1797 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
6 | 5 | exbidv 1798 | . 2 ⊢ (𝑧 = 𝐴 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
7 | ax-sep 4054 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | |
8 | 1, 6, 7 | vtocl 2743 | 1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 ax-sep 4054 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: inex1 4070 bj-d0clsepcl 13294 |
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