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Theorem mpjao3dan 1213
Description: Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
mpjao3dan.1  |-  ( (
ph  /\  ps )  ->  ch )
mpjao3dan.2  |-  ( (
ph  /\  th )  ->  ch )
mpjao3dan.3  |-  ( (
ph  /\  ta )  ->  ch )
mpjao3dan.4  |-  ( ph  ->  ( ps  \/  th  \/  ta ) )
Assertion
Ref Expression
mpjao3dan  |-  ( ph  ->  ch )

Proof of Theorem mpjao3dan
StepHypRef Expression
1 mpjao3dan.1 . . 3  |-  ( (
ph  /\  ps )  ->  ch )
2 mpjao3dan.2 . . 3  |-  ( (
ph  /\  th )  ->  ch )
31, 2jaodan 721 . 2  |-  ( (
ph  /\  ( ps  \/  th ) )  ->  ch )
4 mpjao3dan.3 . 2  |-  ( (
ph  /\  ta )  ->  ch )
5 mpjao3dan.4 . . 3  |-  ( ph  ->  ( ps  \/  th  \/  ta ) )
6 df-3or 897 . . 3  |-  ( ( ps  \/  th  \/  ta )  <->  ( ( ps  \/  th )  \/ 
ta ) )
75, 6sylib 131 . 2  |-  ( ph  ->  ( ( ps  \/  th )  \/  ta )
)
83, 4, 7mpjaodan 722 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    \/ w3o 895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-3or 897
This theorem is referenced by:  wetriext  4329  nntri3  6106  nntri2or2  6107  caucvgprlemnkj  6822  caucvgprlemnbj  6823  caucvgprprlemnkj  6848  caucvgprprlemnbj  6849  caucvgsr  6944  addmodlteq  9348  divalglemeunn  10233  divalglemex  10234  divalglemeuneg  10235  divalg  10236
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