Theorem equeucl 2028

Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2012.) Curried (exported) form of equtr2 2031. (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 2027	. 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 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by:  equtr2  2031  sbequ1  2243  ax13lem1  2374  ax13lem2  2376  bj-ax6elem2  34775  wl-ax13lem1  35592