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Theorem dveeq2 1928
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveeq2  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Distinct variable group:    x, z

Proof of Theorem dveeq2
StepHypRef Expression
1 ax-17 1628 . 2  |-  ( z  =  w  ->  A. x  z  =  w )
2 ax-17 1628 . 2  |-  ( z  =  y  ->  A. w  z  =  y )
3 equequ2 1830 . 2  |-  ( w  =  y  ->  (
z  =  w  <->  z  =  y ) )
41, 2, 3dvelimfALT 1853 1  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  nd5  1931  ax11v2  1935  sbal1  2086  copsexg  4145  axpowndlem3  8101
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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