MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bicom Unicode version

Theorem bicom 193
Description: Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
bicom  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )

Proof of Theorem bicom
StepHypRef Expression
1 bicom1 192 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps 
<-> 
ph ) )
2 bicom1 192 . 2  |-  ( ( ps  <->  ph )  ->  ( ph 
<->  ps ) )
31, 2impbii 182 1  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178
This theorem is referenced by:  bicomd  194  bibi1i  307  bibi1d  312  con2bi  320  ibibr  334  bibif  337  nbbn  349  pm5.17  863  biluk  904  bigolden  906  xorcom  1303  falbitru  1348  3impexpbicom  1363  mtp-xor  1530  exists1  2202  eqcom  2255  abeq1  2355  ssequn1  3255  axpow3  4085  isocnv  5679  qextlt  10408  qextle  10409  rpnnen2  12378  odd2np1  12461  nrmmetd  17929  cvmlift2lem12  23016  wl-bibi2d  24090  oriso  24380  nn0prpw  25405  bicomdd  25871  compneOLD  26810  3impexpbicomVD  27323  bnj926  27488
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator