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Theorem dvdemo1 4209
Description: Demonstration of a theorem (scheme) that requires (meta)variables  x and  y to be distinct, but no others. It bundles the theorem schemes  E. x ( x  =  y  ->  x  e.  x ) and  E. x ( x  =  y  ->  y  e.  x ). Compare dvdemo2 4210. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
Assertion
Ref Expression
dvdemo1  |-  E. x
( x  =  y  ->  z  e.  x
)
Distinct variable group:    x, y

Proof of Theorem dvdemo1
StepHypRef Expression
1 dtru 4200 . . 3  |-  -.  A. x  x  =  y
2 exnal 1562 . . 3  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
31, 2mpbir 202 . 2  |-  E. x  -.  x  =  y
4 pm2.21 102 . . 3  |-  ( -.  x  =  y  -> 
( x  =  y  ->  z  e.  x
) )
54eximi 1564 . 2  |-  ( E. x  -.  x  =  y  ->  E. x
( x  =  y  ->  z  e.  x
) )
63, 5ax-mp 10 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-nul 4150  ax-pow 4187
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533
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