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

Theorem 3eqtrri 2141
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 2136 . 2 𝐴 = 𝐶
4 3eqtri.3 . 2 𝐶 = 𝐷
53, 4eqtr2i 2137 1 𝐷 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1314
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 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108
This theorem is referenced by:  resindm  4829  dfdm2  5041  cofunex2g  5976  df1st2  6082  df2nd2  6083  enq0enq  7203  dfn2  8944  9p1e10  9138  0.999...  11241
  Copyright terms: Public domain W3C validator