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Theorem eqtr 2133
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 2122 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimpar 293 1 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314
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 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108
This theorem is referenced by:  eqtr2  2134  eqtr3  2135  sylan9eq  2168  eqvinc  2780  eqvincg  2781  uneqdifeqim  3416  preqsn  3670  dtruex  4442  relresfld  5036  relcoi1  5038  eqer  6427  xpider  6466  addlsub  8096  bj-findis  12979
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