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Theorem jcab 835
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 445 . . . 4  |-  ( ( ps  /\  ch )  ->  ps )
21imim2i 15 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
3 simpr 449 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 15 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
52, 4jca 520 . 2  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
6 pm3.43 834 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
75, 6impbii 182 1  |-  ( (
ph  ->  ( ps  /\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360
This theorem is referenced by:  ordi  836  pm4.76  838  pm5.44  879  2eu4  2228  ssconb  3311  ssin  3393  tfr3  6411  isprm2  12761  lgsquad2lem2  20593  ostthlem2  20772  2reu4a  27347  pclclN  29348
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362
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