MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveeq2 Unicode version

Theorem dveeq2 1882
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)
Assertion
Ref Expression
dveeq2  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Distinct variable group:    x, z

Proof of Theorem dveeq2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equequ2 1651 . 2  |-  ( w  =  y  ->  (
z  =  w  <->  z  =  y ) )
21dvelimv 1881 1  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1529
This theorem is referenced by:  ax10  1886  ax9  1891  ax11v2  1934  sbal1  2067  copsexg  4254  axpowndlem3  8223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1531
  Copyright terms: Public domain W3C validator