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Theorem dvdemo1 4359
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 4360. ("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 4350 . . 3  |-  -.  A. x  x  =  y
2 exnal 1580 . . 3  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
31, 2mpbir 201 . 2  |-  E. x  -.  x  =  y
4 pm2.21 102 . 2  |-  ( -.  x  =  y  -> 
( x  =  y  ->  z  e.  x
) )
53, 4eximii 1584 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-nul 4298  ax-pow 4337
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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