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Theorem en3lplem1VD 37213
Description: Virtual deduction proof of en3lplem1 8129. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en3lplem1VD  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  E. y ( y  e.  { A ,  B ,  C }  /\  y  e.  x
) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem en3lplem1VD
StepHypRef Expression
1 idn1 36916 . . . . . . 7  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ).
2 simp3 1007 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
31, 2e1a 36978 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  A ).
4 tpid3g 4115 . . . . . 6  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4e1a 36978 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  { A ,  B ,  C } ).
6 idn2 36964 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  x  =  A ).
7 eleq2 2496 . . . . . . 7  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
87biimprd 226 . . . . . 6  |-  ( x  =  A  ->  ( C  e.  A  ->  C  e.  x ) )
96, 3, 8e21 37091 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  C  e.  x ).
10 pm3.2 448 . . . . 5  |-  ( C  e.  { A ,  B ,  C }  ->  ( C  e.  x  ->  ( C  e.  { A ,  B ,  C }  /\  C  e.  x ) ) )
115, 9, 10e12 37085 . . . 4  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  ( C  e.  { A ,  B ,  C }  /\  C  e.  x
) ).
12 elex22 3093 . . . 4  |-  ( ( C  e.  { A ,  B ,  C }  /\  C  e.  x
)  ->  E. y
( y  e.  { A ,  B ,  C }  /\  y  e.  x ) )
1311, 12e2 36982 . . 3  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ).
1413in2 36956 . 2  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( x  =  A  ->  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ) ).
1514in1 36913 1  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  E. y ( y  e.  { A ,  B ,  C }  /\  y  e.  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {ctp 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-sn 3999  df-pr 4001  df-tp 4003  df-vd1 36912  df-vd2 36920
This theorem is referenced by:  en3lplem2VD  37214
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