 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqtr Structured version   GIF version

Theorem eqtr 2035
 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 = B B = 𝐶) → A = 𝐶)

Proof of Theorem eqtr
StepHypRef Expression
1 eqeq1 2024 . 2 (A = B → (A = 𝐶B = 𝐶))
21biimpar 281 1 ((A = B B = 𝐶) → A = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226 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 1312  ax-gen 1314  ax-4 1377  ax-17 1396  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-cleq 2011 This theorem is referenced by:  eqtr2  2036  eqtr3  2037  sylan9eq  2070  eqvinc  2640  eqvincg  2641  uneqdifeqim  3281  preqsn  3516  dtruex  4217  relresfld  4770  relcoi1  4772  eqer  6045  xpiderm  6084  bj-findis  7336
 Copyright terms: Public domain W3C validator