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Theorem imbibi 252
Description: The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
Assertion
Ref Expression
imbibi  |-  ( ( ( ph  ->  ps ) 
<->  ch )  ->  ( ph  ->  ( ps  <->  ch )
) )

Proof of Theorem imbibi
StepHypRef Expression
1 pm5.4 249 . . 3  |-  ( (
ph  ->  ( ph  ->  ps ) )  <->  ( ph  ->  ps ) )
2 imbi2 237 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ch )  ->  (
( ph  ->  ( ph  ->  ps ) )  <->  ( ph  ->  ch ) ) )
31, 2bitr3id 194 . 2  |-  ( ( ( ph  ->  ps ) 
<->  ch )  ->  (
( ph  ->  ps )  <->  (
ph  ->  ch ) ) )
43pm5.74rd 183 1  |-  ( ( ( ph  ->  ps ) 
<->  ch )  ->  ( ph  ->  ( ps  <->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> 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:  snssg  3725
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