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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 3143 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
4 | ssrab 4082 | . 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 2105 ∀wral 3058 {crab 3432 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rab 3433 df-ss 3979 |
This theorem is referenced by: mndind 18853 symggen 19502 ablfac1eu 20107 lspsolvlem 21161 prdsxmslem2 24557 ovolicc2lem4 25568 abelth2 26500 perfectlem2 27288 umgrres1lem 29341 upgrres1 29344 nsgmgc 33419 nsgqusf1olem2 33421 nsgqusf1olem3 33422 cvmlift2lem11 35297 bj-rabtrAUTO 36914 mapdrvallem3 41628 idomsubgmo 43181 nadd2rabtr 43373 k0004ss2 44141 liminfvalxr 45738 smflimlem4 46729 perfectALTVlem2 47646 |
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