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Theorem imim21b 356
Description: Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
Assertion
Ref Expression
imim21b  |-  ( ( ps  ->  ph )  -> 
( ( ( ph  ->  ch )  ->  ( ps  ->  th ) )  <->  ( ps  ->  ( ch  ->  th )
) ) )

Proof of Theorem imim21b
StepHypRef Expression
1 bi2.04 350 . 2  |-  ( ( ( ph  ->  ch )  ->  ( ps  ->  th ) )  <->  ( ps  ->  ( ( ph  ->  ch )  ->  th )
) )
2 pm5.5 326 . . . . 5  |-  ( ph  ->  ( ( ph  ->  ch )  <->  ch ) )
32imbi1d 308 . . . 4  |-  ( ph  ->  ( ( ( ph  ->  ch )  ->  th )  <->  ( ch  ->  th )
) )
43imim2i 13 . . 3  |-  ( ( ps  ->  ph )  -> 
( ps  ->  (
( ( ph  ->  ch )  ->  th )  <->  ( ch  ->  th )
) ) )
54pm5.74d 238 . 2  |-  ( ( ps  ->  ph )  -> 
( ( ps  ->  ( ( ph  ->  ch )  ->  th ) )  <->  ( ps  ->  ( ch  ->  th )
) ) )
61, 5syl5bb 248 1  |-  ( ( ps  ->  ph )  -> 
( ( ( ph  ->  ch )  ->  ( ps  ->  th ) )  <->  ( ps  ->  ( ch  ->  th )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
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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