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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 3434 | . 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | |
2 | vex 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | abbii 2807 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | 1, 4 | eqtr4i 2766 | 1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 {crab 3433 Vcvv 3478 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 |
This theorem is referenced by: notab 4320 intmin2 4980 euen1 9066 dfttrcl2 9762 cardf2 9981 hsmex2 10471 fmla0 35367 fmla0xp 35368 fmla0disjsuc 35383 imageval 35912 rmxyelqirrOLD 42899 dfrcl2 43664 |
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