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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 3147 | . 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | |
2 | vex 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 533 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | abbii 2886 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | 1, 4 | eqtr4i 2847 | 1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-rab 3147 df-v 3496 |
This theorem is referenced by: notab 4273 intmin2 4903 euen1 8579 cardf2 9372 hsmex2 9855 fmla0 32629 fmla0xp 32630 fmla0disjsuc 32645 imageval 33391 rmxyelqirr 39527 dfrcl2 40039 |
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