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Theorem sbc3org 28295
Description: sbcorg 3036 with a 3-disjuncts. This proof is sbc3orgVD 28627 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 3036 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )
) )
2 df-3or 935 . . . . 5  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
32bicomi 193 . . . 4  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  (
ph  \/  ps  \/  ch ) )
43sbcbiiOLD 3047 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch ) 
<-> 
[. A  /  x ]. ( ph  \/  ps  \/  ch ) ) )
5 sbcorg 3036 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps ) 
<->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
65orbi1d 683 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
71, 4, 63bitr3d 274 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) )
8 df-3or 935 . 2  |-  ( (
[. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )
)
97, 8syl6bbr 254 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 176    \/ wo 357    \/ w3o 933    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbcoreleleq  28298  sbcoreleleqVD  28635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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