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Theorem mpan10 443
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 253 . . . 4  |-  ( ( ch  /\  ph )  <->  (
ph  /\  ch )
)
21anbi2i 430 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ch  /\  ph ) )  <->  ( ( ph  ->  ps )  /\  ( ph  /\  ch )
) )
3 anass 381 . . 3  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  <->  ( ( ph  ->  ps )  /\  ( ch  /\  ph )
) )
4 anass 381 . . 3  |-  ( ( ( ( ph  ->  ps )  /\  ph )  /\  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  /\  ch )
) )
52, 3, 43bitr4i 201 . 2  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  <->  ( (
( ph  ->  ps )  /\  ph )  /\  ch ) )
6 id 19 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
76imp 115 . . 3  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
87anim1i 323 . 2  |-  ( ( ( ( ph  ->  ps )  /\  ph )  /\  ch )  ->  ( ps  /\  ch ) )
95, 8sylbi 114 1  |-  ( ( ( ( ph  ->  ps )  /\  ch )  /\  ph )  ->  ( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
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