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Mirrors > Home > ILE Home > Th. List > clelab | GIF version |
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
clelab | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2164 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | 1 | anbi2i 457 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)) |
3 | 2 | exbii 1605 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)) |
4 | df-clel 2173 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
5 | nfv 1528 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝜑) | |
6 | nfv 1528 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 | |
7 | nfs1v 1939 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | 6, 7 | nfan 1565 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑) |
9 | eqeq1 2184 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
10 | sbequ12 1771 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
11 | 9, 10 | anbi12d 473 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))) |
12 | 5, 8, 11 | cbvex 1756 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)) |
13 | 3, 4, 12 | 3bitr4i 212 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 [wsb 1762 ∈ wcel 2148 {cab 2163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 |
This theorem is referenced by: elrabi 2890 |
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