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Theorem nfeqf2 2069
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 1671 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1929 . . 3  |-  F/ x F/ x  z  =  y
3 axc9lem2 2068 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
4 axc9lem1 2030 . . . . 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 1990 . . . 4  |-  ( F/ x  z  =  y  <-> 
( E. x  z  =  y  ->  A. x  z  =  y )
)
75, 6sylibr 214 . . 3  |-  ( -.  x  =  y  ->  F/ x  z  =  y )
82, 7exlimi 1942 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  z  =  y )
91, 8sylbir 215 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1405   E.wex 1635   F/wnf 1639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-12 1880  ax-13 2028
This theorem depends on definitions:  df-bi 187  df-an 371  df-ex 1636  df-nf 1640
This theorem is referenced by:  dveeq2  2070  nfeqf1  2071  sbal1  2230  copsexg  4678  axrepndlem1  9001  axpowndlem2  9007  axpowndlem3  9008  wl-equsb3  31384  wl-sbcom2d-lem1  31389  wl-mo2dnae  31399
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