Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcoreleleq Structured version   Unicode version

Theorem sbcoreleleq 32385
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 32739. (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 3398 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
2 sbcel1v 3396 . . . 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 3393 . . 3  |-  ( A  e.  V  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
5 3orbi123 32360 . . . 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 32301 . . 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 32381 . 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 972    = wceq 1379    e. wcel 1767   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  tratrb  32386  tratrbVD  32741
  Copyright terms: Public domain W3C validator