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Theorem dveel1 2210
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel1  |-  ( -. 
A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z )
)
Distinct variable group:    x, z

Proof of Theorem dveel1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elequ1 1905 . 2  |-  ( w  =  y  ->  (
w  e.  z  <->  y  e.  z ) )
21dvelimv 2183 1  |-  ( -. 
A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  distel  30500
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