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Mirrors > Home > ILE Home > Th. List > rabab | GIF version |
Description: A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
rabab | ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2451 | . 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | |
2 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 301 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | abbii 2280 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | 1, 4 | eqtr4i 2188 | 1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 {crab 2446 Vcvv 2721 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-rab 2451 df-v 2723 |
This theorem is referenced by: notab 3387 intmin2 3844 euen1 6759 bj-omind 13651 |
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