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Mirrors > Home > MPE Home > Th. List > abssdv | Structured version Visualization version GIF version |
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) |
Ref | Expression |
---|---|
abssdv.1 | ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) |
Ref | Expression |
---|---|
abssdv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abssdv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) | |
2 | 1 | alrimiv 1842 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴)) |
3 | abss 3634 | . 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | sylibr 223 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 |
This theorem is referenced by: dfopif 4337 fmpt 6289 opabex2 6997 eroprf 7732 cfslb2n 8973 rankcf 9478 genpv 9700 genpdm 9703 fimaxre3 10849 supadd 10868 supmul 10872 hashfacen 13095 hashf1lem1 13096 hashf1lem2 13097 mertenslem2 14456 4sqlem11 15497 symgbas 17623 lss1d 18784 lspsn 18823 lpval 20753 lpsscls 20755 ptuni2 21189 ptbasfi 21194 prdstopn 21241 xkopt 21268 tgpconcompeqg 21725 metrest 22139 mbfeqalem 23215 limcfval 23442 nmosetre 27003 nmopsetretALT 28106 nmfnsetre 28120 sigaclcuni 29508 bnj849 30249 deranglem 30402 derangsn 30406 liness 31422 mblfinlem3 32618 ismblfin 32620 itg2addnclem 32631 areacirclem2 32671 sdclem2 32708 sdclem1 32709 ismtyval 32769 heibor1lem 32778 heibor1 32779 pmapglbx 34073 eldiophb 36338 hbtlem2 36713 upbdrech 38460 |
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