Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elex22VD Structured version   Visualization version   Unicode version

Theorem elex22VD 37245
Description: Virtual deduction proof of elex22 3061. (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 36955 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( A  e.  B  /\  A  e.  C ) ).
2 simpl 459 . . . . 5  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  B )
31, 2e1a 37017 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  B ).
4 elisset 3059 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
53, 4e1a 37017 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x  x  =  A ).
6 idn2 37003 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  =  A ).
7 eleq1a 2526 . . . . . . . 8  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
83, 6, 7e12 37121 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  B ).
9 simpr 463 . . . . . . . . 9  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  C )
101, 9e1a 37017 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  C ).
11 eleq1a 2526 . . . . . . . 8  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
1210, 6, 11e12 37121 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  C ).
13 pm3.2 449 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  C  -> 
( x  e.  B  /\  x  e.  C
) ) )
148, 12, 13e22 37061 . . . . . 6  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  ( x  e.  B  /\  x  e.  C ) ).
1514in2 36995 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( x  =  A  -> 
( x  e.  B  /\  x  e.  C
) ) ).
1615gen11 37006 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) ).
17 exim 1708 . . . 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 37017 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) ).
19 pm2.27 40 . . 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 37078 . 2  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x ( x  e.  B  /\  x  e.  C ) ).
2120in1 36952 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 371   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-12 1935  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3049  df-vd1 36951  df-vd2 36959
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator