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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqelbid | Structured version Visualization version GIF version |
Description: A variable elimination law for equality within a given set 𝐴. See equvel 2449. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
Ref | Expression |
---|---|
eqelbid.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
eqelbid.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
Ref | Expression |
---|---|
eqelbid | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵)) | |
2 | eqeq1 2730 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐶 ↔ 𝐵 = 𝐶)) | |
3 | 1, 2 | bibi12d 345 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ (𝐵 = 𝐵 ↔ 𝐵 = 𝐶))) |
4 | eqid 2726 | . . . . . 6 ⊢ 𝐵 = 𝐵 | |
5 | 4 | tbt 369 | . . . . 5 ⊢ (𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ↔ 𝐵 = 𝐵)) |
6 | bicom 221 | . . . . 5 ⊢ ((𝐵 = 𝐶 ↔ 𝐵 = 𝐵) ↔ (𝐵 = 𝐵 ↔ 𝐵 = 𝐶)) | |
7 | 5, 6 | bitri 275 | . . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 = 𝐵 ↔ 𝐵 = 𝐶)) |
8 | 3, 7 | bitr4di 289 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶)) |
9 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) → ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) | |
10 | eqelbid.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) → 𝐵 ∈ 𝐴) |
12 | 8, 9, 11 | rspcdva 3607 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) → 𝐵 = 𝐶) |
13 | simplr 766 | . . . 4 ⊢ (((𝜑 ∧ 𝐵 = 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
14 | 13 | eqeq2d 2737 | . . 3 ⊢ (((𝜑 ∧ 𝐵 = 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) |
15 | 14 | ralrimiva 3140 | . 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶)) |
16 | 12, 15 | impbida 798 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 |
This theorem is referenced by: ply1moneq 33168 |
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