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Theorem sbcoreleleq 36306
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 36670. (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 sbcel2gv 3336 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
2 sbcel1v 3335 . . . 4  |-  ( [. A  /  y ]. y  e.  x  <->  A  e.  x
)
32a1i 11 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) )
4 eqsbc3r 3333 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
5 3orbi123 36278 . . . 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 36219 . . 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 ) ) ) ) )
71, 3, 4, 6syl3c 60 . 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 ) ) )
8 sbc3or 36299 . 2  |-  ( [. A  /  y ]. (
x  e.  y  \/  y  e.  x  \/  x  =  y )  <-> 
( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y
) )
97, 8syl6rbbr 264 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ w3o 973    = wceq 1405    e. wcel 1842   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3060  df-sbc 3277
This theorem is referenced by:  tratrb  36307  tratrbVD  36672
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