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Theorem tpid3gVD 29028
Description: Virtual deduction proof of tpid3g 3921. (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 28788 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
2 3mix3 1129 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
31, 2e2 28806 . . . . . . . . 9  |-  (. A  e.  B ,. x  =  A  ->.  ( x  =  C  \/  x  =  D  \/  x  =  A ) ).
4 abid 2426 . . . . . . . . 9  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4e2bir 28808 . . . . . . . 8  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  {
x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ).
6 dftp2 3856 . . . . . . . . 9  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2502 . . . . . . . 8  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7e2bir 28808 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  { C ,  D ,  A } ).
9 eleq1 2498 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
109biimpd 200 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  ->  A  e.  { C ,  D ,  A }
) )
111, 8, 10e22 28846 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  A  e.  { C ,  D ,  A } ).
1211in2 28780 . . . . 5  |-  (. A  e.  B  ->.  ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
1312gen11 28791 . . . 4  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
14 19.23v 1915 . . . 4  |-  ( A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } )  <->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1513, 14e1bi 28804 . . 3  |-  (. A  e.  B  ->.  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) ).
16 idn1 28739 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
17 elisset 2968 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
1816, 17e1_ 28802 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
19 id 21 . . 3  |-  ( ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } )  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
2015, 18, 19e11 28863 . 2  |-  (. A  e.  B  ->.  A  e.  { C ,  D ,  A } ).
2120in1 28736 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 936   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   {ctp 3818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823  df-tp 3824  df-vd1 28735  df-vd2 28744
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