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Theorem en3lplem1VD 28935
Description: Virtual deduction proof of en3lplem1 7432. (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 28641 . . . . . . 7  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ).
2 simp3 957 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
31, 2e1_ 28704 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  A ).
4 tpid3g 3754 . . . . . 6  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4e1_ 28704 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  C  e.  { A ,  B ,  C } ).
6 idn2 28690 . . . . . 6  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  x  =  A ).
7 eleq2 2357 . . . . . . 7  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
87biimprd 214 . . . . . 6  |-  ( x  =  A  ->  ( C  e.  A  ->  C  e.  x ) )
96, 3, 8e21 28819 . . . . 5  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  C  e.  x ).
10 pm3.2 434 . . . . 5  |-  ( C  e.  { A ,  B ,  C }  ->  ( C  e.  x  ->  ( C  e.  { A ,  B ,  C }  /\  C  e.  x ) ) )
115, 9, 10e12 28813 . . . 4  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  ( C  e.  { A ,  B ,  C }  /\  C  e.  x
) ).
12 elex22 2812 . . . 4  |-  ( ( C  e.  { A ,  B ,  C }  /\  C  e.  x
)  ->  E. y
( y  e.  { A ,  B ,  C }  /\  y  e.  x ) )
1311, 12e2 28708 . . 3  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ,. x  =  A  ->.  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ).
1413in2 28682 . 2  |-  (. ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->.  ( x  =  A  ->  E. y ( y  e. 
{ A ,  B ,  C }  /\  y  e.  x ) ) ).
1514in1 28638 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {ctp 3655
This theorem is referenced by:  en3lplem2VD  28936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660  df-tp 3661  df-vd1 28637  df-vd2 28646
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