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Theorem sbcoreleleq 28597
Description: Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28951. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcoreleleq  |-  ( A  e.  V  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
)  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    A( x)    V( x, y)

Proof of Theorem sbcoreleleq
StepHypRef Expression
1 sbc3org 28594 . 2  |-  ( A  e.  V  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
)  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) )
2 sbcel2gv 3064 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
3 sbcel1gv 3063 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) )
4 eqsbc3r 3061 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
5 3orbi123 28572 . . . 4  |-  ( ( ( [. A  / 
y ]. x  e.  y  <-> 
x  e.  A )  /\  ( [. A  /  y ]. y  e.  x  <->  A  e.  x
)  /\  ( [. A  /  y ]. x  =  y  <->  x  =  A
) )  ->  (
( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y
)  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
653impexpbicomi 1358 . . 3  |-  ( (
[. A  /  y ]. x  e.  y  <->  x  e.  A )  -> 
( ( [. A  /  y ]. y  e.  x  <->  A  e.  x
)  ->  ( ( [. A  /  y ]. x  =  y  <->  x  =  A )  -> 
( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ) ) )
72, 3, 4, 6syl3c 57 . 2  |-  ( A  e.  V  ->  (
( x  e.  A  \/  A  e.  x  \/  x  =  A
)  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) )
81, 7bitr4d 247 1  |-  ( A  e.  V  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
)  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ w3o 933    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  tratrb  28598  tratrbVD  28953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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