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Mirrors > Home > NFE 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 3943 | . . . 4 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x)) | |
2 | simpr 447 | . . . . . . . 8 ⊢ ((A ⊆ x ∧ φ) → φ) | |
3 | ancr 532 | . . . . . . . 8 ⊢ ((φ → A ⊆ x) → (φ → (A ⊆ x ∧ φ))) | |
4 | 2, 3 | impbid2 195 | . . . . . . 7 ⊢ ((φ → A ⊆ x) → ((A ⊆ x ∧ φ) ↔ φ)) |
5 | 4 | imbi1d 308 | . . . . . 6 ⊢ ((φ → A ⊆ x) → (((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x))) |
6 | 5 | alimi 1559 | . . . . 5 ⊢ (∀x(φ → A ⊆ x) → ∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x))) |
7 | albi 1564 | . . . . 5 ⊢ (∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) | |
8 | 6, 7 | syl 15 | . . . 4 ⊢ (∀x(φ → A ⊆ x) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) |
9 | 1, 8 | sylbi 187 | . . 3 ⊢ (A ⊆ ∩{x ∣ φ} → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x))) |
10 | vex 2862 | . . . 4 ⊢ y ∈ V | |
11 | 10 | elintab 3937 | . . 3 ⊢ (y ∈ ∩{x ∣ (A ⊆ x ∧ φ)} ↔ ∀x((A ⊆ x ∧ φ) → y ∈ x)) |
12 | 10 | elintab 3937 | . . 3 ⊢ (y ∈ ∩{x ∣ φ} ↔ ∀x(φ → y ∈ x)) |
13 | 9, 11, 12 | 3bitr4g 279 | . 2 ⊢ (A ⊆ ∩{x ∣ φ} → (y ∈ ∩{x ∣ (A ⊆ x ∧ φ)} ↔ y ∈ ∩{x ∣ φ})) |
14 | 13 | eqrdv 2351 | 1 ⊢ (A ⊆ ∩{x ∣ φ} → ∩{x ∣ (A ⊆ x ∧ φ)} = ∩{x ∣ φ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 ⊆ wss 3257 ∩cint 3926 |
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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-int 3927 |
This theorem is referenced by: (None) |
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