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Theorem 3orbi123i 1186
Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
Hypotheses
Ref Expression
bi3.1  |-  ( ph  <->  ps )
bi3.2  |-  ( ch  <->  th )
bi3.3  |-  ( ta  <->  et )
Assertion
Ref Expression
3orbi123i  |-  ( (
ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) )

Proof of Theorem 3orbi123i
StepHypRef Expression
1 bi3.1 . . . 4  |-  ( ph  <->  ps )
2 bi3.2 . . . 4  |-  ( ch  <->  th )
31, 2orbi12i 521 . . 3  |-  ( (
ph  \/  ch )  <->  ( ps  \/  th )
)
4 bi3.3 . . 3  |-  ( ta  <->  et )
53, 4orbi12i 521 . 2  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et )
)
6 df-3or 974 . 2  |-  ( (
ph  \/  ch  \/  ta )  <->  ( ( ph  \/  ch )  \/  ta ) )
7 df-3or 974 . 2  |-  ( ( ps  \/  th  \/  et )  <->  ( ( ps  \/  th )  \/  et ) )
85, 6, 73bitr4i 277 1  |-  ( (
ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    \/ w3o 972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 370  df-3or 974
This theorem is referenced by:  cadcomb  1463  ne3anior  2783  wecmpep  4880  cnvso  5552  sorpss  6584  ordon  6617  soxp  6912  dford2  8054  axlowdimlem6  24377  elxrge02  27788  brtp  29396  socnv  29412  dfon2  29441  sltsolem1  29645
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