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Theorem nfeqf2 2146
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 1710 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1992 . . 3  |-  F/ x F/ x  z  =  y
3 axc9lem2 2144 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
4 axc9lem1 2104 . . . . 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 2052 . . . 4  |-  ( F/ x  z  =  y  <-> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
75, 6sylibr 217 . . 3  |-  ( -.  x  =  y  ->  F/ x  z  =  y )
82, 7exlimi 2006 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  z  =  y )
91, 8sylbir 218 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1453   E.wex 1674   F/wnf 1678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  dveeq2  2147  nfeqf1  2148  sbal1  2300  copsexg  4704  axrepndlem1  9048  axpowndlem2  9054  axpowndlem3  9055  wl-equsb3  31930  wl-sbcom2d-lem1  31935  wl-mo2df  31945  wl-eudf  31947  wl-euequ1f  31949
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