Theorem equtrr 2051

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 2050	. 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
2	1	com12 33	1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 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 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  equeuclr  2052  equequ2  2055  ax12v2  2214  2ax6elem  2501  axprlem3OLD  5399  wl-spae  38204  ax12eq  39743  sn-axprlem3  43017  ax6e2eq  45294  ax6e2eqVD  45643