MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bicom1 Structured version   Unicode version

Theorem bicom1 203
Description: Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
Assertion
Ref Expression
bicom1  |-  ( (
ph 
<->  ps )  ->  ( ps 
<-> 
ph ) )

Proof of Theorem bicom1
StepHypRef Expression
1 biimpr 202 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
2 biimp 197 . 2  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
31, 2impbid 194 1  |-  ( (
ph 
<->  ps )  ->  ( ps 
<-> 
ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 189
This theorem is referenced by:  bicom  204  bicomi  206  bj-con3thALT  31154  rp-fakenanass  36123  frege55aid  36363  frege55lem2a  36365  bisaiaisb  38258  confun4  38287  confun5  38288
  Copyright terms: Public domain W3C validator