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Theorem jcab 833
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 443 . . . 4  |-  ( ( ps  /\  ch )  ->  ps )
21imim2i 13 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
3 simpr 447 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 13 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
52, 4jca 518 . 2  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
6 pm3.43 832 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
75, 6impbii 180 1  |-  ( (
ph  ->  ( ps  /\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  ordi  834  pm4.76  836  pm5.44  877  2eu4  2226  ssconb  3309  ssin  3391  tfr3  6415  isprm2  12766  lgsquad2lem2  20598  ostthlem2  20777  2reu4a  27967  pclclN  30080
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  df-an 360
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