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Theorem nfeqf2 2100
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 1693 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1958 . . 3  |-  F/ x F/ x  z  =  y
3 axc9lem2 2098 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
4 axc9lem1 2059 . . . . 5  |-  ( -.  x  =  y  -> 
( z  =  y  ->  A. x  z  =  y ) )
53, 4syld 45 . . . 4  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
6 nf2 2020 . . . 4  |-  ( F/ x  z  =  y  <-> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
75, 6sylibr 215 . . 3  |-  ( -.  x  =  y  ->  F/ x  z  =  y )
82, 7exlimi 1972 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  z  =  y )
91, 8sylbir 216 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   E.wex 1657   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  dveeq2  2101  nfeqf1  2102  sbal1  2259  copsexg  4706  axrepndlem1  9024  axpowndlem2  9030  axpowndlem3  9031  wl-equsb3  31848  wl-sbcom2d-lem1  31853  wl-mo2df  31863  wl-eudf  31865  wl-euequ1f  31867
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