Theorem equeucl 2026

Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2010.) Curried (exported) form of equtr2 2029. (Contributed by BJ, 11-Apr-2021.)

Assertion

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

Proof of Theorem equeucl

Step	Hyp	Ref	Expression
1		equeuclr 2025	. 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
2	1	com12 32	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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  equtr2  2029  sbequ1  2256  ax13lem1  2379  ax13lem2  2381  bj-ax6elem2  36906  wl-ax13lem1  37743