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Theorem dvdemo2 4211
Description: Demonstration of a theorem (scheme) that requires (meta)variables  x and  z to be distinct, but no others. It bundles the theorem schemes  E. x ( x  =  x  -> 
z  e.  x ) and 
E. x ( x  =  y  ->  y  e.  x ). Compare dvdemo1 4210. (Contributed by NM, 1-Dec-2006.)
Assertion
Ref Expression
dvdemo2  |-  E. x
( x  =  y  ->  z  e.  x
)
Distinct variable group:    x, z

Proof of Theorem dvdemo2
StepHypRef Expression
1 el 4192 . 2  |-  E. x  z  e.  x
2 ax-1 7 . . 3  |-  ( z  e.  x  ->  (
x  =  y  -> 
z  e.  x ) )
32eximi 1564 . 2  |-  ( E. x  z  e.  x  ->  E. x ( x  =  y  ->  z  e.  x ) )
41, 3ax-mp 10 1  |-  E. x
( x  =  y  ->  z  e.  x
)
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1529    = wceq 1624    e. wcel 1685
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-pow 4188
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533
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