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Theorem sbc3org 28327
Description: sbcorg 3166 with a 3-disjuncts. This proof is sbc3orgVD 28672 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbc3org  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )

Proof of Theorem sbc3org
StepHypRef Expression
1 sbcorg 3166 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )
) )
2 df-3or 937 . . . . 5  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
32bicomi 194 . . . 4  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
43sbcbiiOLD 3177 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<-> 
[. A  /  x ]. ( ph  \/  ps  \/  ch ) ) )
5 sbcorg 3166 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
65orbi1d 684 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
71, 4, 63bitr3d 275 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
8 df-3or 937 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
97, 8syl6bbr 255 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    \/ w3o 935    e. wcel 1721   [.wsbc 3121
This theorem is referenced by:  sbcoreleleq  28330  sbcoreleleqVD  28680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122
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