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Theorem dveeq1 1987
Description: Quantifier introduction when one pair of variables is distinct. Revised to be independent of dvelimv 2017. (Contributed by NM, 2-Jan-2002.) (Revised by Wolf Lammen, 27-Feb-2018.)
Assertion
Ref Expression
dveeq1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Distinct variable group:    x, z

Proof of Theorem dveeq1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax12 1985 . . 3  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
2 ax12 1985 . . 3  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
31, 2ax12olem3 1974 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
43nfrd 1775 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:  ax10lem2  1989  ax10  1991  sbal2  2184
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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