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Theorem nfeqf2 2001
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 1619 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1838 . . 3  |-  F/ x F/ x  z  =  y
3 axc9lem2 2000 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
4 axc9lem1 1957 . . . . 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 1900 . . . 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 1850 . 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 1368   E.wex 1587   F/wnf 1590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  dveeq2  2002  nfeqf1  2003  sbal1  2180  copsexg  4687  axpowndlem2  8877  axpowndlem3  8879  wl-equsb3  28551  wl-sbcom2d-lem1  28556  wl-mo2dnae  28566
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