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Theorem en3lplem1VD 37302
Description: Virtual deduction proof of en3lplem1 8137. (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 37012 . . . . . . 7  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ).
2 simp3 1032 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
31, 2e1a 37074 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  A ).
4 tpid3g 4078 . . . . . 6  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4e1a 37074 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  { A ,  B ,  C } ).
6 idn2 37060 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  x  =  A ).
7 eleq2 2538 . . . . . . 7  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
87biimprd 231 . . . . . 6  |-  ( x  =  A  ->  ( C  e.  A  ->  C  e.  x ) )
96, 3, 8e21 37180 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  C  e.  x ).
10 pm3.2 454 . . . . 5  |-  ( C  e.  { A ,  B ,  C }  ->  ( C  e.  x  ->  ( C  e.  { A ,  B ,  C }  /\  C  e.  x ) ) )
115, 9, 10e12 37174 . . . 4  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  ( C  e.  { A ,  B ,  C }  /\  C  e.  x
) ).
12 elex22 3045 . . . 4  |-  ( ( C  e.  { A ,  B ,  C }  /\  C  e.  x
)  ->  E. y
( y  e.  { A ,  B ,  C }  /\  y  e.  x ) )
1311, 12e2 37078 . . 3  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ).
1413in2 37052 . 2  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( x  =  A  ->  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ) ).
1514in1 37009 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 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {ctp 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962  df-tp 3964  df-vd1 37008  df-vd2 37016
This theorem is referenced by:  en3lplem2VD  37303
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