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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviunvw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbviunvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
| cbviunvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| cbviunvw2 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbviunvw2.2 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 2 | cbviunvw2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
| 3 | 2 | eleq2d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
| 4 | 1, 3 | cbvrexvw2 36206 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
| 5 | 4 | abbii 2808 | . 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷} |
| 6 | df-iun 4991 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
| 7 | df-iun 4991 | . 2 ⊢ ∪ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷} | |
| 8 | 5, 6, 7 | 3eqtr4i 2774 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2713 ∃wrex 3069 ∪ ciun 4989 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 df-iun 4991 |
| This theorem is referenced by: (None) |
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