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Theorem dveel1 1896
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 1419 . 2  |-  ( w  e.  z  ->  A. x  w  e.  z )
2 ax-17 1419 . 2  |-  ( y  e.  z  ->  A. w  y  e.  z )
3 elequ1 1600 . 2  |-  ( w  =  y  ->  (
w  e.  z  <->  y  e.  z ) )
41, 2, 3dvelimf 1891 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 1241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by: (None)
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