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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviindavw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and domain in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbviindavw2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) |
| cbviindavw2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| cbviindavw2 | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbviindavw2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) | |
| 2 | 1 | eleq2d 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
| 3 | cbviindavw2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
| 4 | 2, 3 | cbvraldva2 3347 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)) |
| 5 | 4 | abbidv 2807 | . 2 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}) |
| 6 | df-iin 4992 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
| 7 | df-iin 4992 | . 2 ⊢ ∩ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷} | |
| 8 | 5, 6, 7 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 ∀wral 3060 ∩ ciin 4990 |
| 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-ral 3061 df-iin 4992 |
| This theorem is referenced by: (None) |
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