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Theorem dveeq2 2080
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 1700 . 2  |-  ( w  =  y  ->  (
z  =  w  <->  z  =  y ) )
21dvelimv 2077 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 1550
This theorem is referenced by:  ax11v2  2081  ax11v2OLD  2082  sbal1  2209  copsexg  4471  axpowndlem3  8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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