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Theorem eqtr 2057
 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 2046 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimpar 281 1 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 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:  eqtr2  2058  eqtr3  2059  sylan9eq  2092  eqvinc  2667  eqvincg  2668  uneqdifeqim  3308  preqsn  3546  dtruex  4283  relresfld  4847  relcoi1  4849  eqer  6138  xpiderm  6177  bj-findis  10104
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