ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equtr2 Unicode version

Theorem equtr2 1687
Description: A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
equtr2  |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )

Proof of Theorem equtr2
StepHypRef Expression
1 equtrr 1686 . . 3  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
21equcoms 1684 . 2  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
32impcom 124 1  |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-gen 1425  ax-ie2 1470  ax-8 1482  ax-17 1506  ax-i9 1510
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  mo23  2038  euequ1  2092
  Copyright terms: Public domain W3C validator