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Theorem eqtr 2158
 Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
Assertion
Ref Expression
eqtr ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)

Proof of Theorem eqtr
StepHypRef Expression
1 eqeq1 2147 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimpar 295 1 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133 This theorem is referenced by:  eqtr2  2159  eqtr3  2160  sylan9eq  2193  eqvinc  2813  eqvincg  2814  uneqdifeqim  3454  preqsn  3711  dtruex  4484  relresfld  5079  relcoi1  5081  eqer  6472  xpider  6511  addlsub  8183  bj-findis  13392
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