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Theorem jcab 834
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 444 . . . 4  |-  ( ( ps  /\  ch )  ->  ps )
21imim2i 14 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
3 simpr 448 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 14 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
52, 4jca 519 . 2  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
6 pm3.43 833 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
75, 6impbii 181 1  |-  ( (
ph  ->  ( ps  /\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359
This theorem is referenced by:  ordi  835  pm4.76  837  pm5.44  878  2eu4  2363  ssconb  3467  ssin  3550  tfr3  6646  isprm2  13070  lgsquad2lem2  21126  ostthlem2  21305  2reu4a  27876  pclclN  30419
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 178  df-an 361
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