Theorem equtrr 2021

Description: A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Assertion

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

Proof of Theorem equtrr

Step	Hyp	Ref	Expression
1		equtr 2020	. 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
2	1	com12 32	1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  equeuclr  2022  equequ2  2025  ax12v2  2179  2ax6elem  2475  axprlem3OLD  5428  wl-spae  37522  ax12eq  38942  sn-axprlem3  42256  ax6e2eq  44577  ax6e2eqVD  44927