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Theorem nfeqf2 2014
Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.)
Assertion
Ref Expression
nfeqf2  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
Distinct variable group:    x, z

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1628 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1847 . . 3  |-  F/ x F/ x  z  =  y
3 axc9lem2 2013 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
4 axc9lem1 1970 . . . . 5  |-  ( -.  x  =  y  -> 
( z  =  y  ->  A. x  z  =  y ) )
53, 4syld 44 . . . 4  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
6 nf2 1909 . . . 4  |-  ( F/ x  z  =  y  <-> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
75, 6sylibr 212 . . 3  |-  ( -.  x  =  y  ->  F/ x  z  =  y )
82, 7exlimi 1859 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  z  =  y )
91, 8sylbir 213 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1377   E.wex 1596   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  dveeq2  2015  nfeqf1  2016  sbal1  2193  copsexg  4738  axpowndlem2  8985  axpowndlem3  8987  wl-equsb3  29931  wl-sbcom2d-lem1  29936  wl-mo2dnae  29946
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