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Theorem dveeq1-o 2221
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 1976 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Distinct variable group:    x, z

Proof of Theorem dveeq1-o
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1623 . 2  |-  ( w  =  z  ->  A. x  w  =  z )
2 ax-17 1623 . 2  |-  ( y  =  z  ->  A. w  y  =  z )
3 equequ1 1691 . 2  |-  ( w  =  y  ->  (
w  =  z  <->  y  =  z ) )
41, 2, 3dvelimf-o 2214 1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
This theorem is referenced by:  ax11inda2ALT  2232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-4 2169  ax-5o 2170  ax-6o 2171  ax-10o 2173  ax-12o 2176
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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