ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3eqtrri GIF version

Theorem 3eqtrri 2113
Description: An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
3eqtri.1 𝐴 = 𝐵
3eqtri.2 𝐵 = 𝐶
3eqtri.3 𝐶 = 𝐷
Assertion
Ref Expression
3eqtrri 𝐷 = 𝐴

Proof of Theorem 3eqtrri
StepHypRef Expression
1 3eqtri.1 . . 3 𝐴 = 𝐵
2 3eqtri.2 . . 3 𝐵 = 𝐶
31, 2eqtri 2108 . 2 𝐴 = 𝐶
4 3eqtri.3 . 2 𝐶 = 𝐷
53, 4eqtr2i 2109 1 𝐷 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081
This theorem is referenced by:  resindm  4754  dfdm2  4965  cofunex2g  5883  df1st2  5984  df2nd2  5985  enq0enq  6988  dfn2  8684  9p1e10  8877  0.999...  10911
  Copyright terms: Public domain W3C validator