Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2949 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
4 | ssrab 3643 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓)) | |
5 | 1, 3, 4 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ 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-ral 2901 df-rab 2905 df-in 3547 df-ss 3554 |
This theorem is referenced by: mrcmndind 17189 symggen 17713 ablfac1eu 18295 lspsolvlem 18963 prdsxmslem2 22144 ovolicc2lem4 23095 abelth2 24000 perfectlem2 24755 cvmlift2lem11 30549 bj-rabtrAUTO 32121 mapdrvallem3 35953 idomsubgmo 36795 k0004ss2 37470 smflimlem4 39660 perfectALTVlem2 40165 umgrres1lem 40529 upgrres1 40532 |
Copyright terms: Public domain | W3C validator |