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Theorem dvdemo1 4428
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 4429. ("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 4419 . . 3  |-  -.  A. x  x  =  y
2 exnal 1584 . . 3  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
31, 2mpbir 202 . 2  |-  E. x  -.  x  =  y
4 pm2.21 103 . 2  |-  ( -.  x  =  y  -> 
( x  =  y  ->  z  e.  x
) )
53, 4eximii 1588 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550   E.wex 1551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-nul 4363  ax-pow 4406
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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