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| Mirrors > Home > MPE Home > Th. List > rabab | Structured version Visualization version 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 3436 | . 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | |
| 2 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) | 
| 4 | 3 | abbii 2808 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | 
| 5 | 1, 4 | eqtr4i 2767 | 1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 {crab 3435 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 | 
| This theorem is referenced by: notab 4313 intmin2 4974 euen1 9068 dfttrcl2 9765 cardf2 9984 hsmex2 10474 fmla0 35388 fmla0xp 35389 fmla0disjsuc 35404 imageval 35932 rmxyelqirrOLD 42927 dfrcl2 43692 | 
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