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Theorem tpid3gVD 33375
Description: Virtual deduction proof of tpid3g 4130. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tpid3gVD  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )

Proof of Theorem tpid3gVD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 idn2 33132 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
2 3mix3 1168 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
31, 2e2 33150 . . . . . . . . 9  |-  (. A  e.  B ,. x  =  A  ->.  ( x  =  C  \/  x  =  D  \/  x  =  A ) ).
4 abid 2430 . . . . . . . . 9  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4e2bir 33152 . . . . . . . 8  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  {
x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ).
6 dftp2 4060 . . . . . . . . 9  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2521 . . . . . . . 8  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7e2bir 33152 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  { C ,  D ,  A } ).
9 eleq1 2515 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
109biimpd 207 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  ->  A  e.  { C ,  D ,  A }
) )
111, 8, 10e22 33190 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  A  e.  { C ,  D ,  A } ).
1211in2 33124 . . . . 5  |-  (. A  e.  B  ->.  ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
1312gen11 33135 . . . 4  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
14 19.23v 1747 . . . 4  |-  ( A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } )  <->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1513, 14e1bi 33148 . . 3  |-  (. A  e.  B  ->.  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) ).
16 idn1 33084 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
17 elisset 3106 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
1816, 17e1a 33146 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
19 id 22 . . 3  |-  ( ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } )  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
2015, 18, 19e11 33207 . 2  |-  (. A  e.  B  ->.  A  e.  { C ,  D ,  A } ).
2120in1 33081 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 973   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428   {ctp 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466  df-sn 4015  df-pr 4017  df-tp 4019  df-vd1 33080  df-vd2 33088
This theorem is referenced by: (None)
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