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Mirrors > Home > MPE Home > Th. List > Mathboxes > elunif | Structured version Visualization version GIF version |
Description: A version of eluni 4833 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
elunif.1 | ⊢ Ⅎ𝑥𝐴 |
elunif.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
elunif | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 4833 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) | |
2 | elunif.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
4 | 2, 3 | nfel 2989 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 |
5 | elunif.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 3, 5 | nfel 2989 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfan 1891 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) |
8 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) | |
9 | eleq2w 2893 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
10 | eleq1w 2892 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | anbi12d 630 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
12 | 7, 8, 11 | cbvexv1 2353 | . 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
13 | 1, 12 | bitri 276 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 Ⅎwnfc 2958 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-uni 4831 |
This theorem is referenced by: eluni2f 41246 stoweidlem46 42208 stoweidlem57 42219 |
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