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| Mirrors > Home > MPE Home > Th. List > ssrabdv | Structured version Visualization version GIF version | ||
| Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.) |
| Ref | Expression |
|---|---|
| ssrabdv.1 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| ssrabdv.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓) |
| Ref | Expression |
|---|---|
| ssrabdv | ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssrabdv.1 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 2 | ssrabdv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓) | |
| 3 | 2 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
| 4 | ssrab 4073 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓)) | |
| 5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-ss 3968 |
| This theorem is referenced by: mndind 18841 symggen 19488 ablfac1eu 20093 lspsolvlem 21144 prdsxmslem2 24542 ovolicc2lem4 25555 abelth2 26486 perfectlem2 27274 umgrres1lem 29327 upgrres1 29330 nsgmgc 33440 nsgqusf1olem2 33442 nsgqusf1olem3 33443 cvmlift2lem11 35318 bj-rabtrAUTO 36933 mapdrvallem3 41648 idomsubgmo 43205 nadd2rabtr 43397 k0004ss2 44165 liminfvalxr 45798 smflimlem4 46789 perfectALTVlem2 47709 |
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