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

Theorem bicom 139
Description: Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)
Assertion
Ref Expression
bicom  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )

Proof of Theorem bicom
StepHypRef Expression
1 bicom1 130 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps 
<-> 
ph ) )
2 bicom1 130 . 2  |-  ( ( ps  <->  ph )  ->  ( ph 
<->  ps ) )
31, 2impbii 125 1  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bicomd  140  bibi1i  227  bibi1d  232  ibibr  245  bibif  688  con2bidc  861  con2biddc  866  pm5.17dc  890  bigolden  940  nbbndc  1373  bilukdc  1375  falbitru  1396  3impexpbicom  1415  exists1  2096  eqcom  2142  abeq1  2250  necon2abiddc  2375  necon2bbiddc  2376  necon4bbiddc  2383  ssequn1  3250  axpow3  4108  isocnv  5719  suplocsrlem  7639  uzennn  10239  bezoutlemle  11730
  Copyright terms: Public domain W3C validator