Theorem equtr2 2062

Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2059. (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 2059	. 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
2	1	imp 412	1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  nfeqf  2413  mo3  2592  mo4  2594  madurid  22989  dchrisumlema  27845  funpartfun  36885  wl-mo3t  38700  discsubc  50383