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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 3114 | . 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} | |
2 | vex 3440 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 531 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | abbii 2861 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | 1, 4 | eqtr4i 2822 | 1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∈ wcel 2081 {cab 2775 {crab 3109 Vcvv 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-rab 3114 df-v 3439 |
This theorem is referenced by: notab 4193 intmin2 4809 euen1 8427 cardf2 9218 hsmex2 9701 fmla0 32237 fmla0xp 32238 fmla0disjsuc 32253 imageval 33000 rmxyelqirr 38992 dfrcl2 39504 |
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