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Mirrors > Home > MPE Home > Th. List > equeuclr | Structured version Visualization version GIF version |
Description: Commuted version of equeucl 2031 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.) |
Ref | Expression |
---|---|
equeuclr | ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtrr 2029 | . 2 ⊢ (𝑧 = 𝑥 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) | |
2 | 1 | equcoms 2027 | 1 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) |
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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: equeucl 2031 equequ2 2033 ax13b 2039 aevlem0 2059 sbequivvOLD 2334 axc15 2444 equviniOLD 2478 sbequiOLD 2534 sbequiALT 2596 euequ 2683 sn-dtru 39131 |
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