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Theorem dvdemo1 4635
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 4636. ("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 4594 . . 3  |-  -.  A. x  x  =  y
2 exnal 1699 . . 3  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
31, 2mpbir 213 . 2  |-  E. x  -.  x  =  y
4 pm2.21 112 . 2  |-  ( -.  x  =  y  -> 
( x  =  y  ->  z  e.  x
) )
53, 4eximii 1709 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442   E.wex 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-nul 4534  ax-pow 4581
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
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