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Theorem tpid3gVD 28618
Description: Virtual deduction proof of tpid3g 3741. (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 28385 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
2 3mix3 1126 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
31, 2e2 28403 . . . . . . . . 9  |-  (. A  e.  B ,. x  =  A  ->.  ( x  =  C  \/  x  =  D  \/  x  =  A ) ).
4 abid 2271 . . . . . . . . 9  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4e2bir 28405 . . . . . . . 8  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  {
x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ).
6 dftp2 3679 . . . . . . . . 9  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2347 . . . . . . . 8  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7e2bir 28405 . . . . . . 7  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  { C ,  D ,  A } ).
9 eleq1 2343 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
109biimpd 198 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  ->  A  e.  { C ,  D ,  A }
) )
111, 8, 10e22 28443 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  A  e.  { C ,  D ,  A } ).
1211in2 28377 . . . . 5  |-  (. A  e.  B  ->.  ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
1312gen11 28388 . . . 4  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } ) ).
14 19.23v 1832 . . . 4  |-  ( A. x ( x  =  A  ->  A  e.  { C ,  D ,  A } )  <->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1513, 14e1bi 28401 . . 3  |-  (. A  e.  B  ->.  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A }
) ).
16 idn1 28342 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
17 elisset 2798 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
1816, 17e1_ 28399 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
19 id 19 . . 3  |-  ( ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } )  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
2015, 18, 19e11 28460 . 2  |-  (. A  e.  B  ->.  A  e.  { C ,  D ,  A } ).
2120in1 28339 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 933   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   {ctp 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648  df-vd1 28338  df-vd2 28347
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