Theorem equtr2 2025

Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2022. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)

Assertion

Ref	Expression
equtr2	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)

Proof of Theorem equtr2

Step	Hyp	Ref	Expression
1		equeucl 2022	. 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
2	1	imp 406	1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 395

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:  nfeqf  2385  mo3  2563  mo4  2565  madurid  22651  dchrisumlema  27533  funpartfun  35945  wl-mo3t  37578