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Theorem dveel2 2151
Description: Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel2  |-  ( -. 
A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
)
Distinct variable group:    x, z

Proof of Theorem dveel2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1519 . 2  |-  ( z  e.  w  ->  A. x  z  e.  w )
2 ax-17 1519 . 2  |-  ( z  e.  y  ->  A. w  z  e.  y )
3 elequ2 2146 . 2  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
41, 2, 3dvelimf 2008 1  |-  ( -. 
A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by: (None)
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