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Theorem eqtr2 2058
 Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
eqtr2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)

Proof of Theorem eqtr2
StepHypRef Expression
1 eqcom 2042 . 2 (𝐴 = 𝐵𝐵 = 𝐴)
2 eqtr 2057 . 2 ((𝐵 = 𝐴𝐴 = 𝐶) → 𝐵 = 𝐶)
31, 2sylanb 268 1 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033 This theorem is referenced by:  eqvinc  2667  eqvincg  2668  moop2  3988  reusv3i  4191  relop  4486  fliftfun  5436  th3qlem1  6208  enq0ref  6531  enq0tr  6532  genpdisj  6621
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