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Theorem wl-eudf 31945
Description: Version of df-eu 2313 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
Hypotheses
Ref Expression
wl-eudf.1  |-  F/ x ph
wl-eudf.2  |-  F/ y
ph
wl-eudf.3  |-  ( ph  ->  -.  A. x  x  =  y )
wl-eudf.4  |-  ( ph  ->  F/ y ps )
Assertion
Ref Expression
wl-eudf  |-  ( ph  ->  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y
) ) )

Proof of Theorem wl-eudf
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 df-eu 2313 . 2  |-  ( E! x ps  <->  E. u A. x ( ps  <->  x  =  u ) )
2 wl-eudf.2 . . 3  |-  F/ y
ph
3 wl-eudf.1 . . . 4  |-  F/ x ph
4 wl-eudf.4 . . . . 5  |-  ( ph  ->  F/ y ps )
5 wl-eudf.3 . . . . . 6  |-  ( ph  ->  -.  A. x  x  =  y )
6 nfeqf1 2147 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y  x  =  u )
76naecoms 2157 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ y  x  =  u )
85, 7syl 17 . . . . 5  |-  ( ph  ->  F/ y  x  =  u )
94, 8nfbid 2026 . . . 4  |-  ( ph  ->  F/ y ( ps  <->  x  =  u ) )
103, 9nfald 2044 . . 3  |-  ( ph  ->  F/ y A. x
( ps  <->  x  =  u ) )
11 nfnae 2162 . . . . . . 7  |-  F/ x  -.  A. x  x  =  y
12 nfeqf2 2145 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/ x  u  =  y )
1311, 12nfan1 2020 . . . . . 6  |-  F/ x
( -.  A. x  x  =  y  /\  u  =  y )
14 equequ2 1878 . . . . . . . 8  |-  ( u  =  y  ->  (
x  =  u  <->  x  =  y ) )
1514bibi2d 324 . . . . . . 7  |-  ( u  =  y  ->  (
( ps  <->  x  =  u )  <->  ( ps  <->  x  =  y ) ) )
1615adantl 472 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  u  =  y )  -> 
( ( ps  <->  x  =  u )  <->  ( ps  <->  x  =  y ) ) )
1713, 16albid 1973 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  u  =  y )  -> 
( A. x ( ps  <->  x  =  u
)  <->  A. x ( ps  <->  x  =  y ) ) )
185, 17sylan 478 . . . 4  |-  ( (
ph  /\  u  =  y )  ->  ( A. x ( ps  <->  x  =  u )  <->  A. x
( ps  <->  x  =  y ) ) )
1918ex 440 . . 3  |-  ( ph  ->  ( u  =  y  ->  ( A. x
( ps  <->  x  =  u )  <->  A. x
( ps  <->  x  =  y ) ) ) )
202, 10, 19cbvexd 2129 . 2  |-  ( ph  ->  ( E. u A. x ( ps  <->  x  =  u )  <->  E. y A. x ( ps  <->  x  =  y ) ) )
211, 20syl5bb 265 1  |-  ( ph  ->  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673   F/wnf 1677   E!weu 2309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-eu 2313
This theorem is referenced by:  wl-eutf  31946
  Copyright terms: Public domain W3C validator