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Mirrors > Home > MPE Home > Th. List > Mathboxes > elunif | Structured version Visualization version GIF version |
Description: A version of eluni 4834 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 4834 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) | |
2 | elunif.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2976 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
4 | 2, 3 | nfel 2991 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 |
5 | elunif.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 3, 5 | nfel 2991 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) |
8 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) | |
9 | eleq2w 2895 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
10 | eleq1w 2894 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
12 | 7, 8, 11 | cbvexv1 2361 | . 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
13 | 1, 12 | bitri 277 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1779 ∈ wcel 2113 Ⅎwnfc 2960 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-uni 4832 |
This theorem is referenced by: eluni2f 41443 stoweidlem46 42405 stoweidlem57 42416 |
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