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Theorem dveeq1 2022
Description: Quantifier introduction when one pair of variables is distinct. Revised to be independent of dvelimv 2075. (Contributed by NM, 2-Jan-2002.) (Revised by Wolf Lammen, 29-Apr-2018.)
Assertion
Ref Expression
dveeq1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Distinct variable group:    x, z

Proof of Theorem dveeq1
StepHypRef Expression
1 ax12 2020 . . 3  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
21ax12olem3 2008 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
32nfrd 1780 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 1550
This theorem is referenced by:  ax10lem2  2024  ax10  2026  sbal2  2213
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
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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