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Theorem elex22VD 34020
Description: Virtual deduction proof of elex22 3060. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex22VD  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elex22VD
StepHypRef Expression
1 idn1 33726 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( A  e.  B  /\  A  e.  C ) ).
2 simpl 455 . . . . 5  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  B )
31, 2e1a 33788 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  B ).
4 elisset 3058 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
53, 4e1a 33788 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x  x  =  A ).
6 idn2 33774 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  =  A ).
7 eleq1a 2475 . . . . . . . 8  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
83, 6, 7e12 33896 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  B ).
9 simpr 459 . . . . . . . . 9  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  C )
101, 9e1a 33788 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  C ).
11 eleq1a 2475 . . . . . . . 8  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
1210, 6, 11e12 33896 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  C ).
13 pm3.2 445 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  C  -> 
( x  e.  B  /\  x  e.  C
) ) )
148, 12, 13e22 33832 . . . . . 6  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  ( x  e.  B  /\  x  e.  C ) ).
1514in2 33766 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( x  =  A  -> 
( x  e.  B  /\  x  e.  C
) ) ).
1615gen11 33777 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) ).
17 exim 1669 . . . 4  |-  ( A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) )  -> 
( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) )
1816, 17e1a 33788 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) ).
19 pm2.27 39 . . 3  |-  ( E. x  x  =  A  ->  ( ( E. x  x  =  A  ->  E. x ( x  e.  B  /\  x  e.  C ) )  ->  E. x ( x  e.  B  /\  x  e.  C ) ) )
205, 18, 19e11 33849 . 2  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x ( x  e.  B  /\  x  e.  C ) ).
2120in1 33723 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397    = wceq 1399   E.wex 1627    e. wcel 1836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-12 1872  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-sb 1758  df-clab 2378  df-cleq 2384  df-clel 2387  df-v 3049  df-vd1 33722  df-vd2 33730
This theorem is referenced by: (None)
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