Theorem equeuclr 2026

Description: Commuted version of equeucl 2027 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)

Assertion

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

Proof of Theorem equeuclr

Step	Hyp	Ref	Expression
1		equtrr 2025	. 2 ⊢ (𝑧 = 𝑥 → (𝑦 = 𝑧 → 𝑦 = 𝑥))
2	1	equcoms 2023	1 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  equeucl  2027  equequ2  2029  ax13b  2035  aevlem0  2055  sbequivvOLD  2330  axc15  2440  equviniOLD  2474  sbequiOLD  2530  sbequiALT  2592  euequ  2679  sn-dtru  39104