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Mirrors > Home > NFE Home > Th. List > rabss | GIF version |
Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
Ref | Expression |
---|---|
rabss | ⊢ ({x ∈ A ∣ φ} ⊆ B ↔ ∀x ∈ A (φ → x ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2623 | . . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
2 | 1 | sseq1i 3295 | . 2 ⊢ ({x ∈ A ∣ φ} ⊆ B ↔ {x ∣ (x ∈ A ∧ φ)} ⊆ B) |
3 | abss 3335 | . 2 ⊢ ({x ∣ (x ∈ A ∧ φ)} ⊆ B ↔ ∀x((x ∈ A ∧ φ) → x ∈ B)) | |
4 | impexp 433 | . . . 4 ⊢ (((x ∈ A ∧ φ) → x ∈ B) ↔ (x ∈ A → (φ → x ∈ B))) | |
5 | 4 | albii 1566 | . . 3 ⊢ (∀x((x ∈ A ∧ φ) → x ∈ B) ↔ ∀x(x ∈ A → (φ → x ∈ B))) |
6 | df-ral 2619 | . . 3 ⊢ (∀x ∈ A (φ → x ∈ B) ↔ ∀x(x ∈ A → (φ → x ∈ B))) | |
7 | 5, 6 | bitr4i 243 | . 2 ⊢ (∀x((x ∈ A ∧ φ) → x ∈ B) ↔ ∀x ∈ A (φ → x ∈ B)) |
8 | 2, 3, 7 | 3bitri 262 | 1 ⊢ ({x ∈ A ∣ φ} ⊆ B ↔ ∀x ∈ A (φ → x ∈ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 {cab 2339 ∀wral 2614 {crab 2618 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: rabssdv 3346 |
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