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Theorem 3orbi123d 1272
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
Hypotheses
Ref Expression
bi3d.1  |-  ( ph  ->  ( ps  <->  ch )
)
bi3d.2  |-  ( ph  ->  ( th  <->  ta )
)
bi3d.3  |-  ( ph  ->  ( et  <->  ze )
)
Assertion
Ref Expression
3orbi123d  |-  ( ph  ->  ( ( ps  \/  th  \/  et )  <->  ( ch  \/  ta  \/  ze )
) )

Proof of Theorem 3orbi123d
StepHypRef Expression
1 bi3d.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
2 bi3d.2 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
31, 2orbi12d 765 . . 3  |-  ( ph  ->  ( ( ps  \/  th )  <->  ( ch  \/  ta ) ) )
4 bi3d.3 . . 3  |-  ( ph  ->  ( et  <->  ze )
)
53, 4orbi12d 765 . 2  |-  ( ph  ->  ( ( ( ps  \/  th )  \/  et )  <->  ( ( ch  \/  ta )  \/ 
ze ) ) )
6 df-3or 946 . 2  |-  ( ( ps  \/  th  \/  et )  <->  ( ( ps  \/  th )  \/  et ) )
7 df-3or 946 . 2  |-  ( ( ch  \/  ta  \/  ze )  <->  ( ( ch  \/  ta )  \/ 
ze ) )
85, 6, 73bitr4g 222 1  |-  ( ph  ->  ( ( ps  \/  th  \/  et )  <->  ( ch  \/  ta  \/  ze )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 680    \/ w3o 944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116  df-3or 946
This theorem is referenced by:  ordtriexmid  4405  wetriext  4459  nntri3or  6355  tridc  6759  ltsopi  7092  pitri3or  7094  nqtri3or  7168  elz  9007  ztri3or  9048  qtri3or  9960  trilpo  13038
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