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Mirrors > Home > MPE Home > Th. List > rabeqc | Structured version Visualization version GIF version |
Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) |
Ref | Expression |
---|---|
rabeqc.1 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
rabeqc | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3147 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | abeq1 2946 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴)) | |
3 | rabeqc.1 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝜑) | |
4 | 3 | pm4.71i 562 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | bicomi 226 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴) |
6 | 2, 5 | mpgbir 1796 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 |
7 | 1, 6 | eqtri 2844 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 {crab 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rab 3147 |
This theorem is referenced by: 2clwwlk2 28121 numclwwlk3lem2lem 28156 fply1 30926 elnanelprv 32671 |
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