| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iineq12dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.) |
| Ref | Expression |
|---|---|
| iineq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| iineq12dv.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| iineq12dv | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iineq12dv.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | eleq2d 2827 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 3 | 2 | imbi1d 341 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶))) |
| 4 | 3 | ralbidv2 3174 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)) |
| 5 | 4 | abbidv 2808 | . . 3 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}) |
| 6 | df-iin 4994 | . . 3 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
| 7 | df-iin 4994 | . . 3 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | . 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) |
| 9 | iineq12dv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) | |
| 10 | 9 | iineq2dv 5017 | . 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐵 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
| 11 | 8, 10 | eqtrd 2777 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∩ ciin 4992 |
| 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 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-iin 4994 |
| This theorem is referenced by: smflim 46792 |
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