Theorem equeucl 2031

Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2015.) Curried (exported) form of equtr2 2034. (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 2030	. 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 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  equtr2  2034  sbequ1  2249  ax13lem1  2392  ax13lem2  2394  bj-ax6elem2  34000  wl-ax13lem1  34761