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Theorem 3orim123d 1226
Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
Hypotheses
Ref Expression
3anim123d.1  |-  ( ph  ->  ( ps  ->  ch ) )
3anim123d.2  |-  ( ph  ->  ( th  ->  ta ) )
3anim123d.3  |-  ( ph  ->  ( et  ->  ze )
)
Assertion
Ref Expression
3orim123d  |-  ( ph  ->  ( ( ps  \/  th  \/  et )  -> 
( ch  \/  ta  \/  ze ) ) )

Proof of Theorem 3orim123d
StepHypRef Expression
1 3anim123d.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
2 3anim123d.2 . . . 4  |-  ( ph  ->  ( th  ->  ta ) )
31, 2orim12d 710 . . 3  |-  ( ph  ->  ( ( ps  \/  th )  ->  ( ch  \/  ta ) ) )
4 3anim123d.3 . . 3  |-  ( ph  ->  ( et  ->  ze )
)
53, 4orim12d 710 . 2  |-  ( ph  ->  ( ( ( ps  \/  th )  \/  et )  ->  (
( ch  \/  ta )  \/  ze )
) )
6 df-3or 897 . 2  |-  ( ( ps  \/  th  \/  et )  <->  ( ( ps  \/  th )  \/  et ) )
7 df-3or 897 . 2  |-  ( ( ch  \/  ta  \/  ze )  <->  ( ( ch  \/  ta )  \/ 
ze ) )
85, 6, 73imtr4g 198 1  |-  ( ph  ->  ( ( ps  \/  th  \/  et )  -> 
( ch  \/  ta  \/  ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ 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:  ztri3or0  8344
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