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Mirrors > Home > NFE Home > Th. List > iinab | GIF version |
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.) |
Ref | Expression |
---|---|
iinab | ⊢ ∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . . . 4 ⊢ ℲyA | |
2 | nfab1 2491 | . . . 4 ⊢ Ⅎy{y ∣ φ} | |
3 | 1, 2 | nfiin 3996 | . . 3 ⊢ Ⅎy∩x ∈ A {y ∣ φ} |
4 | nfab1 2491 | . . 3 ⊢ Ⅎy{y ∣ ∀x ∈ A φ} | |
5 | 3, 4 | cleqf 2513 | . 2 ⊢ (∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ} ↔ ∀y(y ∈ ∩x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∀x ∈ A φ})) |
6 | abid 2341 | . . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ) | |
7 | 6 | ralbii 2638 | . . 3 ⊢ (∀x ∈ A y ∈ {y ∣ φ} ↔ ∀x ∈ A φ) |
8 | vex 2862 | . . . 4 ⊢ y ∈ V | |
9 | eliin 3974 | . . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A {y ∣ φ} ↔ ∀x ∈ A y ∈ {y ∣ φ})) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (y ∈ ∩x ∈ A {y ∣ φ} ↔ ∀x ∈ A y ∈ {y ∣ φ}) |
11 | abid 2341 | . . 3 ⊢ (y ∈ {y ∣ ∀x ∈ A φ} ↔ ∀x ∈ A φ) | |
12 | 7, 10, 11 | 3bitr4i 268 | . 2 ⊢ (y ∈ ∩x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∀x ∈ A φ}) |
13 | 5, 12 | mpgbir 1550 | 1 ⊢ ∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 {cab 2339 ∀wral 2614 Vcvv 2859 ∩ciin 3970 |
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-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-iin 3972 |
This theorem is referenced by: iinrab 4028 |
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