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

Theorem sbcoreleleqVD 36900
Description: Virtual deduction proof of sbcoreleleq 36537. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1a 36648  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1a 36648  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1a 36648  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 36695  |-  (. A  e.  B  ->.  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
6:1,?: e1a 36648  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
7:5,6: e11 36709  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) ).
qed:7:  |-  ( A  e.  B  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcoreleleqVD  |-  ( A  e.  B  ->  ( [. 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)    B( x, y)

Proof of Theorem sbcoreleleqVD
StepHypRef Expression
1 idn1 36586 . . . . 5  |-  (. A  e.  B  ->.  A  e.  B ).
2 sbcel2gv 3356 . . . . 5  |-  ( A  e.  B  ->  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) )
31, 2e1a 36648 . . . 4  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A
) ).
4 sbcel1gvOLD 36899 . . . . 5  |-  ( A  e.  B  ->  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) )
51, 4e1a 36648 . . . 4  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x
) ).
6 eqsbc3r 3353 . . . . 5  |-  ( A  e.  B  ->  ( [. A  /  y ]. x  =  y  <->  x  =  A ) )
71, 6e1a 36648 . . . 4  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A
) ).
8 3orbi123 36509 . . . . 5  |-  ( ( ( [. 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 ) ) )
983impexpbicomi 36476 . . . 4  |-  ( (
[. 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 ) ) ) ) )
103, 5, 7, 9e111 36695 . . 3  |-  (. A  e.  B  ->.  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
11 sbc3orgOLD 36534 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
)  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) )
121, 11e1a 36648 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  y ]. (
x  e.  y  \/  y  e.  x  \/  x  =  y )  <-> 
( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y
) ) ).
13 biantr 939 . . . 4  |-  ( ( ( [. A  / 
y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) )  /\  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) )  ->  ( [. A  /  y ]. (
x  e.  y  \/  y  e.  x  \/  x  =  y )  <-> 
( x  e.  A  \/  A  e.  x  \/  x  =  A
) ) )
1413expcom 436 . . 3  |-  ( ( ( x  e.  A  \/  A  e.  x  \/  x  =  A
)  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) )  ->  ( ( [. A  /  y ]. (
x  e.  y  \/  y  e.  x  \/  x  =  y )  <-> 
( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y
) )  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
)  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) ) )
1510, 12, 14e11 36709 . 2  |-  (. A  e.  B  ->.  ( [. A  /  y ]. (
x  e.  y  \/  y  e.  x  \/  x  =  y )  <-> 
( x  e.  A  \/  A  e.  x  \/  x  =  A
) ) ).
1615in1 36583 1  |-  ( A  e.  B  ->  ( [. 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 187    \/ w3o 981    = wceq 1437    e. wcel 1867   [.wsbc 3296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-v 3080  df-sbc 3297  df-vd1 36582
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator