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Theorem tpid3gVD 37143
Description: Virtual deduction proof of tpid3g 4113. (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 36895 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
2 3mix3 1177 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
31, 2e2 36913 . . . . . . . . 9  |-  (. A  e.  B ,. x  =  A  ->.  ( x  =  C  \/  x  =  D  \/  x  =  A ) ).
4 abid 2410 . . . . . . . . 9  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4e2bir 36915 . . . . . . . 8  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  {
x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ).
6 dftp2 4044 . . . . . . . . 9  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2501 . . . . . . . 8  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7e2bir 36915 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  { C ,  D ,  A } ).
9 eleq1 2495 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
109biimpd 211 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  ->  A  e.  { C ,  D ,  A }
) )
111, 8, 10e22 36953 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  A  e.  { C ,  D ,  A } ).
1211in2 36887 . . . . 5  |-  (. A  e.  B  ->.  ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
1312gen11 36898 . . . 4  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
14 19.23v 1808 . . . 4  |-  ( A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } )  <->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1513, 14e1bi 36911 . . 3  |-  (. A  e.  B  ->.  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) ).
16 idn1 36847 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
17 elisset 3093 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
1816, 17e1a 36909 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
19 id 23 . . 3  |-  ( ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } )  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
2015, 18, 19e11 36970 . 2  |-  (. A  e.  B  ->.  A  e.  { C ,  D ,  A } ).
2120in1 36844 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 982   A.wal 1436    = wceq 1438   E.wex 1660    e. wcel 1869   {cab 2408   {ctp 4001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-un 3442  df-sn 3998  df-pr 4000  df-tp 4002  df-vd1 36843  df-vd2 36851
This theorem is referenced by: (None)
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