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Mirrors > Home > MPE Home > Th. List > Mathboxes > iineq1i | Structured version Visualization version GIF version |
Description: Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
iineq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
iineq1i | ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iineq1i.1 | . . . . . 6 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2829 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | imbi1i 349 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶)) |
4 | 3 | ralbii2 3085 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
5 | 4 | abbii 2805 | . 2 ⊢ {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} |
6 | df-iin 5002 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
7 | df-iin 5002 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4i 2771 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1535 ∈ wcel 2104 {cab 2710 ∀wral 3057 ∩ ciin 5000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-iin 5002 |
This theorem is referenced by: (None) |
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