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Theorem jcab 545
Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
Assertion
Ref Expression
jcab  |-  ( (
ph  ->  ( ps  /\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch )
) )

Proof of Theorem jcab
StepHypRef Expression
1 simpl 106 . . . 4  |-  ( ( ps  /\  ch )  ->  ps )
21imim2i 12 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
3 simpr 107 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 12 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
52, 4jca 294 . 2  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
6 pm3.43 544 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
75, 6impbii 121 1  |-  ( (
ph  ->  ( ps  /\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  pm4.76  546  pm5.44  845  2eu4  2009  ssconb  3104  ssin  3187  raaan  3355  tfri3  5984
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