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

Theorem eqtr 2073
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 2062 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimpar 285 1 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  eqtr2  2074  eqtr3  2075  sylan9eq  2108  eqvinc  2690  eqvincg  2691  uneqdifeqim  3336  preqsn  3574  dtruex  4311  relresfld  4875  relcoi1  4877  eqer  6169  xpiderm  6208  addlsub  7440  bj-findis  10491
  Copyright terms: Public domain W3C validator