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Theorem dvdemo1 4386
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 4387. ("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 4377 . . 3  |-  -.  A. x  x  =  y
2 exnal 1583 . . 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 1587 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-nul 4325  ax-pow 4364
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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