ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpan10 Unicode version

Theorem mpan10 466
Description: Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
Assertion
Ref Expression
mpan10  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  ->  ( ps  /\  ch ) )

Proof of Theorem mpan10
StepHypRef Expression
1 ancom 264 . . . 4  |-  ( ( ch  /\  ph )  <->  (
ph  /\  ch )
)
21anbi2i 453 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ch  /\  ph ) )  <->  ( ( ph  ->  ps )  /\  ( ph  /\  ch )
) )
3 anass 399 . . 3  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  <->  ( ( ph  ->  ps )  /\  ( ch  /\  ph )
) )
4 anass 399 . . 3  |-  ( ( ( ( ph  ->  ps )  /\  ph )  /\  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  /\  ch )
) )
52, 3, 43bitr4i 211 . 2  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  <->  ( (
( ph  ->  ps )  /\  ph )  /\  ch ) )
6 id 19 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
76imp 123 . . 3  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
87anim1i 338 . 2  |-  ( ( ( ( ph  ->  ps )  /\  ph )  /\  ch )  ->  ( ps  /\  ch ) )
95, 8sylbi 120 1  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  ->  ( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator