Theorem et-equeucl 46832

Description: Alternative proof that equality is left-Euclidean, using ax7 2014 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.)

Assertion

Ref	Expression
et-equeucl	⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))

Proof of Theorem et-equeucl

Step	Hyp	Ref	Expression
1		equid 2010	. . 3 ⊢ 𝑥 = 𝑥
2		ax7 2014	. . . 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 2010	. . . . 5 ⊢ 𝑦 = 𝑦
6		ax7 2014	. . . . . 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 2014	. . . . 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))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)