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Theorem sbc3or 32381
Description: sbcor 3376 with a 3-disjuncts. This proof is sbc3orgVD 32731 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbc3or  |-  ( [. A  /  x ]. ( ph  \/  ps  \/  ch ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )

Proof of Theorem sbc3or
StepHypRef Expression
1 sbcor 3376 . . 3  |-  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  (
[. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) )
2 df-3or 974 . . . . 5  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
32bicomi 202 . . . 4  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
43sbcbii 3391 . . 3  |-  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) )
5 sbcor 3376 . . . 4  |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
65orbi1i 520 . . 3  |-  ( (
[. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) )
71, 4, 63bitr3i 275 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps  \/  ch ) 
<->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
8 df-3or 974 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
97, 8bitr4i 252 1  |-  ( [. A  /  x ]. ( ph  \/  ps  \/  ch ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    \/ w3o 972   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  sbcoreleleq  32385
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