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Theorem wl-euequ1f 31973
Description: euequ1 2325 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
Assertion
Ref Expression
wl-euequ1f  |-  ( -. 
A. x  x  =  y  ->  E! x  x  =  y )

Proof of Theorem wl-euequ1f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1815 . . 3  |-  E. z 
z  =  y
2 nfv 1769 . . . 4  |-  F/ z  -.  A. x  x  =  y
3 nfnae 2167 . . . . 5  |-  F/ x  -.  A. x  x  =  y
4 nfeqf2 2148 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
5 equequ2 1876 . . . . . . 7  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
65equcoms 1872 . . . . . 6  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
76a1i 11 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  <->  x  =  z ) ) )
83, 4, 7alrimdd 1978 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x
( x  =  y  <-> 
x  =  z ) ) )
92, 8eximd 1980 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  E. z A. x
( x  =  y  <-> 
x  =  z ) ) )
101, 9mpi 20 . 2  |-  ( -. 
A. x  x  =  y  ->  E. z A. x ( x  =  y  <->  x  =  z
) )
11 df-eu 2323 . 2  |-  ( E! x  x  =  y  <->  E. z A. x ( x  =  y  <->  x  =  z ) )
1210, 11sylibr 217 1  |-  ( -. 
A. x  x  =  y  ->  E! x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450   E.wex 1671   E!weu 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323
This theorem is referenced by: (None)
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