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Theorem pm5.32d 445
Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
Hypothesis
Ref Expression
pm5.32d.1  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
Assertion
Ref Expression
pm5.32d  |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )

Proof of Theorem pm5.32d
StepHypRef Expression
1 pm5.32d.1 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
2 bi1 117 . . . 4  |-  ( ( ch  <->  th )  ->  ( ch  ->  th ) )
31, 2syl6 33 . . 3  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
43imdistand 443 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( ps 
/\  th ) ) )
5 bi2 129 . . . 4  |-  ( ( ch  <->  th )  ->  ( th  ->  ch ) )
61, 5syl6 33 . . 3  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
76imdistand 443 . 2  |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ps  /\ 
ch ) ) )
84, 7impbid 128 1  |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  pm5.32rd  446  pm5.32da  447  pm5.32  448  anbi2d  459  cbvex2  1894  cores  5042  isoini  5719  mpoeq123  5830  genpassl  7332  genpassu  7333  fzind  9166  btwnz  9170  elfzm11  9871  isprm2  11798  isprm3  11799
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