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Mirrors > Home > MPE Home > Th. List > clelab | Structured version Visualization version GIF version |
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
Ref | Expression |
---|---|
clelab | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2832 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
2 | nfv 1916 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝜑) | |
3 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 | |
4 | nfsab1 2745 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
5 | 3, 4 | nfan 1901 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
6 | eqeq1 2763 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
7 | sbequ12 2251 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | df-clab 2737 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
9 | 7, 8 | bitr4di 293 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})) |
10 | 6, 9 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))) |
11 | 2, 5, 10 | cbvexv1 2352 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) |
12 | 1, 11 | bitr4i 281 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1539 ∃wex 1782 [wsb 2070 ∈ wcel 2112 {cab 2736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 |
This theorem is referenced by: elrabiOLD 3595 bj-csbsnlem 34609 frege55c 40977 spr0nelg 44346 |
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