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Theorem impbid21d 182
Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
Hypotheses
Ref Expression
impbid21d.1  |-  ( ps 
->  ( ch  ->  th )
)
impbid21d.2  |-  ( ph  ->  ( th  ->  ch ) )
Assertion
Ref Expression
impbid21d  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )

Proof of Theorem impbid21d
StepHypRef Expression
1 impbid21d.1 . . 3  |-  ( ps 
->  ( ch  ->  th )
)
21a1i 10 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
3 impbid21d.2 . . 3  |-  ( ph  ->  ( th  ->  ch ) )
43a1d 22 . 2  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
52, 4impbidd 181 1  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
This theorem is referenced by:  impbid  183  pm5.1im  229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177
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