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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 3104 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
4 | ssrab 3821 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓)) | |
5 | 1, 3, 4 | sylanbrc 701 | 1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 ∀wral 3050 {crab 3054 ⊆ wss 3715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rab 3059 df-in 3722 df-ss 3729 |
This theorem is referenced by: mrcmndind 17587 symggen 18110 ablfac1eu 18692 lspsolvlem 19364 prdsxmslem2 22555 ovolicc2lem4 23508 abelth2 24415 perfectlem2 25175 umgrres1lem 26422 upgrres1 26425 cvmlift2lem11 31623 bj-rabtrAUTO 33254 mapdrvallem3 37455 idomsubgmo 38296 k0004ss2 38970 liminfvalxr 40536 smflimlem4 41506 perfectALTVlem2 42159 |
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