Theorem equtrr 2055

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 2054	. 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 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:  equeuclr  2056  equequ2  2059  ax12v2  2218  2ax6elem  2505  axprlem3OLD  5405  wl-spae  38217  ax12eq  39756  sn-axprlem3  43030  ax6e2eq  45307  ax6e2eqVD  45656