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Theorem tpid3gVD 37248
Description: Virtual deduction proof of tpid3g 4090. (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 37003 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
2 3mix3 1180 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
31, 2e2 37021 . . . . . . . . 9  |-  (. A  e.  B ,. x  =  A  ->.  ( x  =  C  \/  x  =  D  \/  x  =  A ) ).
4 abid 2441 . . . . . . . . 9  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4e2bir 37023 . . . . . . . 8  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  {
x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ).
6 dftp2 4020 . . . . . . . . 9  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2523 . . . . . . . 8  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7e2bir 37023 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  { C ,  D ,  A } ).
9 eleq1 2519 . . . . . . . 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 37061 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  A  e.  { C ,  D ,  A } ).
1211in2 36995 . . . . 5  |-  (. A  e.  B  ->.  ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
1312gen11 37006 . . . 4  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
14 19.23v 1820 . . . 4  |-  ( A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } )  <->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1513, 14e1bi 37019 . . 3  |-  (. A  e.  B  ->.  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) ).
16 idn1 36955 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
17 elisset 3059 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
1816, 17e1a 37017 . . 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 37078 . 2  |-  (. A  e.  B  ->.  A  e.  { C ,  D ,  A } ).
2120in1 36952 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 985   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889   {cab 2439   {ctp 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049  df-un 3411  df-sn 3971  df-pr 3973  df-tp 3975  df-vd1 36951  df-vd2 36959
This theorem is referenced by: (None)
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