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Theorem sbcoreleleq 31554
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 31908. (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 3355 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
2 sbcel1v 3353 . . . 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 3350 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
5 3orbi123 31529 . . . 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 31470 . . 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 61 . 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 31550 . 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 964    = wceq 1370    e. wcel 1758   [.wsbc 3288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3074  df-sbc 3289
This theorem is referenced by:  tratrb  31555  tratrbVD  31910
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