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Theorem pm4.87 567
Description: Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
Assertion
Ref Expression
pm4.87  |-  ( ( ( ( ( ph  /\ 
ps )  ->  ch ) 
<->  ( ph  ->  ( ps  ->  ch ) ) )  /\  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ps  ->  ( ph  ->  ch ) ) ) )  /\  ( ( ps 
->  ( ph  ->  ch ) )  <->  ( ( ps  /\  ph )  ->  ch ) ) )

Proof of Theorem pm4.87
StepHypRef Expression
1 impexp 433 . . 3  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch ) ) )
2 bi2.04 350 . . 3  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ps  ->  ( ph  ->  ch ) ) )
31, 2pm3.2i 441 . 2  |-  ( ( ( ( ph  /\  ps )  ->  ch )  <->  (
ph  ->  ( ps  ->  ch ) ) )  /\  ( ( ph  ->  ( ps  ->  ch )
)  <->  ( ps  ->  (
ph  ->  ch ) ) ) )
4 impexp 433 . . 3  |-  ( ( ( ps  /\  ph )  ->  ch )  <->  ( ps  ->  ( ph  ->  ch ) ) )
54bicomi 193 . 2  |-  ( ( ps  ->  ( ph  ->  ch ) )  <->  ( ( ps  /\  ph )  ->  ch ) )
63, 5pm3.2i 441 1  |-  ( ( ( ( ( ph  /\ 
ps )  ->  ch ) 
<->  ( ph  ->  ( ps  ->  ch ) ) )  /\  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ps  ->  ( ph  ->  ch ) ) ) )  /\  ( ( ps 
->  ( ph  ->  ch ) )  <->  ( ( ps  /\  ph )  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
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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