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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 rule). (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 3115 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
4 | ssrab 3829 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓)) | |
5 | 1, 3, 4 | sylanbrc 572 | 1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ∀wral 3061 {crab 3065 ⊆ wss 3723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rab 3070 df-in 3730 df-ss 3737 |
This theorem is referenced by: mrcmndind 17574 symggen 18097 ablfac1eu 18680 lspsolvlem 19356 prdsxmslem2 22554 ovolicc2lem4 23508 abelth2 24416 perfectlem2 25176 umgrres1lem 26425 upgrres1 26428 clwwlknonclwlknonf1olemOLD 27550 cvmlift2lem11 31633 bj-rabtrAUTO 33261 mapdrvallem3 37456 idomsubgmo 38302 k0004ss2 38976 liminfvalxr 40533 smflimlem4 41502 perfectALTVlem2 42159 |
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