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Mathbox for Ender Ting |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > et-equeucl | Structured version Visualization version GIF version |
Description: Alternative proof that equality is left-Euclidean, using ax7 2020 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.) |
Ref | Expression |
---|---|
et-equeucl | ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2016 | . . 3 ⊢ 𝑥 = 𝑥 | |
2 | ax7 2020 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑥 → 𝑧 = 𝑥)) | |
3 | 2 | com12 32 | . . 3 ⊢ (𝑥 = 𝑥 → (𝑥 = 𝑧 → 𝑧 = 𝑥)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝑥 = 𝑧 → 𝑧 = 𝑥) |
5 | equid 2016 | . . . . 5 ⊢ 𝑦 = 𝑦 | |
6 | ax7 2020 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑦 → 𝑧 = 𝑦)) | |
7 | 6 | com12 32 | . . . . 5 ⊢ (𝑦 = 𝑦 → (𝑦 = 𝑧 → 𝑧 = 𝑦)) |
8 | 5, 7 | ax-mp 5 | . . . 4 ⊢ (𝑦 = 𝑧 → 𝑧 = 𝑦) |
9 | ax7 2020 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦)) | |
10 | 9 | com12 32 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 → 𝑥 = 𝑦)) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑧 = 𝑥 → 𝑥 = 𝑦)) |
12 | 11 | com12 32 | . 2 ⊢ (𝑧 = 𝑥 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) |
13 | 4, 12 | syl 17 | 1 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: (None) |
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