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Mirrors > Home > MPE Home > Th. List > rabssdv | Structured version Visualization version GIF version |
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
rabssdv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
rabssdv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabssdv.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵) | |
2 | 1 | 3exp 1264 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵))) |
3 | 2 | ralrimiv 2965 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵)) |
4 | rabss 3679 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | sylibr 224 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-in 3581 df-ss 3588 |
This theorem is referenced by: suppss2 7329 oemapvali 8581 cantnflem1 8586 harval2 8823 zsupss 11777 ramub1lem1 15730 symggen 17890 efgsfo 18152 ablfacrp 18465 ablfac1eu 18472 pgpfac1lem5 18478 ablfaclem3 18486 nrmr0reg 21552 ptcmplem3 21858 abelthlem2 24186 lgamgulmlem1 24755 neibastop2lem 32355 topmeet 32359 cntotbnd 33595 mapdrvallem2 36934 k0004ss1 38449 liminfvalxr 40015 |
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