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Theorem dvdemo1 4182
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 4183. ("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 4173 . . 3  |-  -.  A. x  x  =  y
2 exnal 1572 . . 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 1574 . 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 1532   E.wex 1537    = wceq 1619    e. wcel 1621
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-13 1625  ax-14 1626  ax-17 1628  ax-9 1684  ax-4 1692  ax-nul 4123  ax-pow 4160
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540
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