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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 3182 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
4 | ssrab 4048 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓)) | |
5 | 1, 3, 4 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 {crab 3142 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-in 3942 df-ss 3951 |
This theorem is referenced by: mndind 17986 symggen 18592 ablfac1eu 19189 lspsolvlem 19908 prdsxmslem2 23133 ovolicc2lem4 24115 abelth2 25024 perfectlem2 25800 umgrres1lem 27086 upgrres1 27089 cvmlift2lem11 32555 bj-rabtrAUTO 34245 mapdrvallem3 38776 idomsubgmo 39791 k0004ss2 40495 liminfvalxr 42057 smflimlem4 43044 perfectALTVlem2 43881 |
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