MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm4.14 Unicode version

Theorem pm4.14 563
Description: Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
Assertion
Ref Expression
pm4.14  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ph  /\  -.  ch )  ->  -.  ps ) )

Proof of Theorem pm4.14
StepHypRef Expression
1 con34b 285 . . 3  |-  ( ( ps  ->  ch )  <->  ( -.  ch  ->  -.  ps ) )
21imbi2i 305 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ph  ->  ( -.  ch  ->  -. 
ps ) ) )
3 impexp 435 . 2  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch ) ) )
4 impexp 435 . 2  |-  ( ( ( ph  /\  -.  ch )  ->  -.  ps ) 
<->  ( ph  ->  ( -.  ch  ->  -.  ps )
) )
52, 3, 43bitr4i 270 1  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ph  /\  -.  ch )  ->  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360
This theorem is referenced by:  pm3.37  564  ndvdssub  12601
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
  Copyright terms: Public domain W3C validator