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Theorem sbc3or 36332
Description: sbcor 3323 with a 3-disjuncts. This proof is sbc3orgVD 36694 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 3323 . . 3  |-  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  (
[. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) )
2 df-3or 977 . . . . 5  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
32bicomi 204 . . . 4  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
43sbcbii 3335 . . 3  |-  ( [. A  /  x ]. (
( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) )
5 sbcor 3323 . . . 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 277 . 2  |-  ( [. A  /  x ]. ( ph  \/  ps  \/  ch ) 
<->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
8 df-3or 977 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
97, 8bitr4i 254 1  |-  ( [. A  /  x ]. ( ph  \/  ps  \/  ch ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    \/ wo 368    \/ w3o 975   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-v 3063  df-sbc 3280
This theorem is referenced by:  sbcoreleleq  36339
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