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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviindavw | Structured version Visualization version GIF version |
Description: Change bound variable in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbviindavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviindavw | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviindavw.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) | |
2 | 1 | eleq2d 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶)) |
3 | 2 | cbvraldva 3238 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶)) |
4 | 3 | abbidv 2807 | . 2 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} = {𝑡 ∣ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶}) |
5 | df-iin 4992 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} | |
6 | df-iin 4992 | . 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶} | |
7 | 4, 5, 6 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶) |
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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