Theorem equeucl 2057

Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2041.) Curried (exported) form of equtr2 2060. (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 2056	. 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
2	1	com12 33	1 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))

This proof depends on syntax axioms:   → wi 4

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 1943  ax-6 2000  ax-7 2041

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

This theorem is used by:  equtr2  2060  sbequ1  2287  ax13lem1  2409  ax13lem2  2411  bj-ax6elem2  37330  wl-ax13lem1  38181