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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  1987  sbrbif  1989  abeq2  2313  abid2f  2373  necon4biddc  2450  pm13.183  2910  disj3  3512  euabsn2  3701  a9evsep  4165  inex1  4177  zfpair2  4253  sucel  4456  bdinex1  15797  bj-zfpair2  15808  bj-d0clsepcl  15823
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