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Mirrors > Home > MPE Home > Th. List > elqsg | Structured version Visualization version GIF version |
Description: Closed form of elqs 8199. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Ref | Expression |
---|---|
elqsg | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2799 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅)) | |
2 | 1 | rexbidv 3260 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
3 | df-qs 8145 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
4 | 2, 3 | elab2g 3607 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 [cec 8137 / cqs 8138 |
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-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-qs 8145 |
This theorem is referenced by: elqs 8199 elqsi 8200 elqsecl 8201 ecelqsg 8202 elpi1 23332 eldmqsres 35075 eldmqs1cossres 35424 prtlem11 35533 |
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