ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bibi2i Unicode version

Theorem bibi2i 227
Description: Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
Hypothesis
Ref Expression
bibi.a  |-  ( ph  <->  ps )
Assertion
Ref Expression
bibi2i  |-  ( ( ch  <->  ph )  <->  ( ch  <->  ps ) )

Proof of Theorem bibi2i
StepHypRef Expression
1 id 19 . . 3  |-  ( ( ch  <->  ph )  ->  ( ch 
<-> 
ph ) )
2 bibi.a . . 3  |-  ( ph  <->  ps )
31, 2bitrdi 196 . 2  |-  ( ( ch  <->  ph )  ->  ( ch 
<->  ps ) )
4 id 19 . . 3  |-  ( ( ch  <->  ps )  ->  ( ch 
<->  ps ) )
54, 2bitr4di 198 . 2  |-  ( ( ch  <->  ps )  ->  ( ch 
<-> 
ph ) )
63, 5impbii 126 1  |-  ( ( ch  <->  ph )  <->  ( ch  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  bibi1i  228  bibi12i  229  bibi2d  232  pm4.71r  390  sblbis  1976  sbrbif  1978  abeq2  2302  abid2f  2362  necon4biddc  2439  pm13.183  2898  disj3  3499  euabsn2  3687  a9evsep  4151  inex1  4163  zfpair2  4239  sucel  4441  bdinex1  15391  bj-zfpair2  15402  bj-d0clsepcl  15417
  Copyright terms: Public domain W3C validator