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Theorem sbcoreleleq 36966
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 37319. (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 3315 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
2 sbcel1v 3314 . . . 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 3312 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
5 3orbi123 36938 . . . 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 36905 . . 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 62 . 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 36959 . 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 272 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 189    \/ w3o 1006    = wceq 1452    e. wcel 1904   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256
This theorem is referenced by:  tratrb  36967  tratrbVD  37321
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