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Mirrors > Home > MPE Home > Th. List > elqsg | Structured version Visualization version GIF version |
Description: Closed form of elqs 7799. (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 2626 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅)) | |
2 | 1 | rexbidv 3052 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
3 | df-qs 7748 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
4 | 2, 3 | elab2g 3353 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 [cec 7740 / cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-qs 7748 |
This theorem is referenced by: elqs 7799 elqsi 7800 elqsecl 7801 ecelqsg 7802 elpi1 22845 eldmqsres 34051 prtlem11 34151 |
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