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Theorem dveel1 1973
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 ax-17 1491 . 2  |-  ( w  e.  z  ->  A. x  w  e.  z )
2 ax-17 1491 . 2  |-  ( y  e.  z  ->  A. w  y  e.  z )
3 elequ1 1675 . 2  |-  ( w  =  y  ->  (
w  e.  z  <->  y  e.  z ) )
41, 2, 3dvelimf 1968 1  |-  ( -. 
A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1314
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 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by: (None)
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