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Theorem dveel2 2138
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 1506 . 2  |-  ( z  e.  w  ->  A. x  z  e.  w )
2 ax-17 1506 . 2  |-  ( z  e.  y  ->  A. w  z  e.  y )
3 elequ2 2133 . 2  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
41, 2, 3dvelimf 1995 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 1333
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 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by: (None)
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