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

Theorem 3eqtrrd 2093
Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
3eqtrd.1  |-  ( ph  ->  A  =  B )
3eqtrd.2  |-  ( ph  ->  B  =  C )
3eqtrd.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
3eqtrrd  |-  ( ph  ->  D  =  A )

Proof of Theorem 3eqtrrd
StepHypRef Expression
1 3eqtrd.1 . . 3  |-  ( ph  ->  A  =  B )
2 3eqtrd.2 . . 3  |-  ( ph  ->  B  =  C )
31, 2eqtrd 2088 . 2  |-  ( ph  ->  A  =  C )
4 3eqtrd.3 . 2  |-  ( ph  ->  C  =  D )
53, 4eqtr2d 2089 1  |-  ( ph  ->  D  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  nnanq0  6614  1idprl  6746  1idpru  6747  axcnre  7013  fseq1p1m1  9058  expmulzap  9466  expubnd  9477  subsq  9525  bcm1k  9628  bcpasc  9634  crim  9686  rereb  9691
  Copyright terms: Public domain W3C validator