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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpreq | Structured version Visualization version GIF version |
Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
Ref | Expression |
---|---|
elpreq.1 | ⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵}) |
elpreq.2 | ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵}) |
elpreq.3 | ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴)) |
Ref | Expression |
---|---|
elpreq | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑋 = 𝐴) | |
2 | elpreq.3 | . . . 4 ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴)) | |
3 | 2 | biimpa 475 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑌 = 𝐴) |
4 | 1, 3 | eqtr4d 2769 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑋 = 𝑌) |
5 | elpreq.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵}) | |
6 | elpri 4656 | . . . . 5 ⊢ (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴 ∨ 𝑋 = 𝐵)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 = 𝐴 ∨ 𝑋 = 𝐵)) |
8 | 7 | orcanai 1000 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝐵) |
9 | simpl 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑) | |
10 | 2 | notbid 317 | . . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴)) |
11 | 10 | biimpa 475 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴) |
12 | elpreq.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵}) | |
13 | elpri 4656 | . . . . 5 ⊢ (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴 ∨ 𝑌 = 𝐵)) | |
14 | pm2.53 849 | . . . . 5 ⊢ ((𝑌 = 𝐴 ∨ 𝑌 = 𝐵) → (¬ 𝑌 = 𝐴 → 𝑌 = 𝐵)) | |
15 | 12, 13, 14 | 3syl 18 | . . . 4 ⊢ (𝜑 → (¬ 𝑌 = 𝐴 → 𝑌 = 𝐵)) |
16 | 9, 11, 15 | sylc 65 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑌 = 𝐵) |
17 | 8, 16 | eqtr4d 2769 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌) |
18 | 4, 17 | pm2.61dan 811 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 {cpr 4635 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3952 df-sn 4634 df-pr 4636 |
This theorem is referenced by: indpreima 33858 |
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