Theorem equtrr 2068

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 2067	. 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 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818

This theorem is referenced by:  equeuclr  2069  equequ2  2072  equvinv  2076  equvelv  2078  ax12v2  2162  2ax6elem  2550  wl-spae  33538  wl-ax8clv2  33613  ax12eq  34647  sbeqalbi  39020  ax6e2eq  39192  ax6e2eqVD  39559