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Theorem eqtr 2135
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 2124 . 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 1316
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 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110
This theorem is referenced by:  eqtr2  2136  eqtr3  2137  sylan9eq  2170  eqvinc  2782  eqvincg  2783  uneqdifeqim  3418  preqsn  3672  dtruex  4444  relresfld  5038  relcoi1  5040  eqer  6429  xpider  6468  addlsub  8100  bj-findis  13104
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