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Theorem dveeq1-o 2263
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2021 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 1626 . 2  |-  ( w  =  z  ->  A. x  w  =  z )
2 ax-17 1626 . 2  |-  ( y  =  z  ->  A. w  y  =  z )
3 equequ1 1696 . 2  |-  ( w  =  y  ->  (
w  =  z  <->  y  =  z ) )
41, 2, 3dvelimf-o 2256 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 1549
This theorem is referenced by:  ax11inda2ALT  2274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-4 2211  ax-5o 2212  ax-6o 2213  ax-10o 2215  ax-12o 2218
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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