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Theorem dveeq2-o 2262
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2078 using ax-11o 2219. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq2-o  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Distinct variable group:    x, z

Proof of Theorem dveeq2-o
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1627 . 2  |-  ( z  =  w  ->  A. x  z  =  w )
2 ax-17 1627 . 2  |-  ( z  =  y  ->  A. w  z  =  y )
3 equequ2 1699 . 2  |-  ( w  =  y  ->  (
z  =  w  <->  z  =  y ) )
41, 2, 3dvelimf-o 2258 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:  ax11eq  2271  ax11el  2272  ax11inda  2278  ax11v2-o  2279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-4 2213  ax-5o 2214  ax-6o 2215  ax-10o 2217  ax-12o 2220
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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