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Theorem sbc3orgOLD 31236
Description: sbcorgOLD 3231 with a 3-disjuncts. This proof is sbc3orgVD 31585 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbc3orgOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )

Proof of Theorem sbc3orgOLD
StepHypRef Expression
1 sbcorgOLD 3231 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )
) )
2 df-3or 966 . . . . 5  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
32bicomi 202 . . . 4  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
43sbcbiiOLD 3246 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<-> 
[. A  /  x ]. ( ph  \/  ps  \/  ch ) ) )
5 sbcorgOLD 3231 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
65orbi1d 702 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
71, 4, 63bitr3d 283 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
8 df-3or 966 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
97, 8syl6bbr 263 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    \/ w3o 964    e. wcel 1756   [.wsbc 3185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-v 2973  df-sbc 3186
This theorem is referenced by:  sbcoreleleqVD  31593
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