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Theorem et-equeucl 45187
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.)
Assertion
Ref Expression
et-equeucl (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Proof of Theorem et-equeucl
StepHypRef Expression
1 equid 2016 . . 3 𝑥 = 𝑥
2 ax7 2020 . . . 4 (𝑥 = 𝑧 → (𝑥 = 𝑥𝑧 = 𝑥))
32com12 32 . . 3 (𝑥 = 𝑥 → (𝑥 = 𝑧𝑧 = 𝑥))
41, 3ax-mp 5 . 2 (𝑥 = 𝑧𝑧 = 𝑥)
5 equid 2016 . . . . 5 𝑦 = 𝑦
6 ax7 2020 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = 𝑦𝑧 = 𝑦))
76com12 32 . . . . 5 (𝑦 = 𝑦 → (𝑦 = 𝑧𝑧 = 𝑦))
85, 7ax-mp 5 . . . 4 (𝑦 = 𝑧𝑧 = 𝑦)
9 ax7 2020 . . . . 5 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
109com12 32 . . . 4 (𝑧 = 𝑦 → (𝑧 = 𝑥𝑥 = 𝑦))
118, 10syl 17 . . 3 (𝑦 = 𝑧 → (𝑧 = 𝑥𝑥 = 𝑦))
1211com12 32 . 2 (𝑧 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑦))
134, 12syl 17 1 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 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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