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Mirrors > Home > ILE Home > Th. List > intmin4 | GIF version |
Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.) |
Ref | Expression |
---|---|
intmin4 | ⊢ (A ⊆ ∩ {x ∣ φ} → ∩ {x ∣ (A ⊆ x ∧ φ)} = ∩ {x ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintab 3623 | . . . 4 ⊢ (A ⊆ ∩ {x ∣ φ} ↔ ∀x(φ → A ⊆ x)) | |
2 | simpr 103 | . . . . . . . 8 ⊢ ((A ⊆ x ∧ φ) → φ) | |
3 | ancr 304 | . . . . . . . 8 ⊢ ((φ → A ⊆ x) → (φ → (A ⊆ x ∧ φ))) | |
4 | 2, 3 | impbid2 131 | . . . . . . 7 ⊢ ((φ → A ⊆ x) → ((A ⊆ x ∧ φ) ↔ φ)) |
5 | 4 | imbi1d 220 | . . . . . 6 ⊢ ((φ → A ⊆ x) → (((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x))) |
6 | 5 | alimi 1341 | . . . . 5 ⊢ (∀x(φ → A ⊆ x) → ∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x))) |
7 | albi 1354 | . . . . 5 ⊢ (∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) | |
8 | 6, 7 | syl 14 | . . . 4 ⊢ (∀x(φ → A ⊆ x) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) |
9 | 1, 8 | sylbi 114 | . . 3 ⊢ (A ⊆ ∩ {x ∣ φ} → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) |
10 | vex 2554 | . . . 4 ⊢ y ∈ V | |
11 | 10 | elintab 3617 | . . 3 ⊢ (y ∈ ∩ {x ∣ (A ⊆ x ∧ φ)} ↔ ∀x((A ⊆ x ∧ φ) → y ∈ x)) |
12 | 10 | elintab 3617 | . . 3 ⊢ (y ∈ ∩ {x ∣ φ} ↔ ∀x(φ → y ∈ x)) |
13 | 9, 11, 12 | 3bitr4g 212 | . 2 ⊢ (A ⊆ ∩ {x ∣ φ} → (y ∈ ∩ {x ∣ (A ⊆ x ∧ φ)} ↔ y ∈ ∩ {x ∣ φ})) |
14 | 13 | eqrdv 2035 | 1 ⊢ (A ⊆ ∩ {x ∣ φ} → ∩ {x ∣ (A ⊆ x ∧ φ)} = ∩ {x ∣ φ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 {cab 2023 ⊆ wss 2911 ∩ cint 3606 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-in 2918 df-ss 2925 df-int 3607 |
This theorem is referenced by: (None) |
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