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Theorem wl-mo2df 31943
Description: Version of mo2 2318 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, 11-Aug-2019.)
Hypotheses
Ref Expression
wl-mo2df.1  |-  F/ x ph
wl-mo2df.2  |-  F/ y
ph
wl-mo2df.3  |-  ( ph  ->  -.  A. x  x  =  y )
wl-mo2df.4  |-  ( ph  ->  F/ y ps )
Assertion
Ref Expression
wl-mo2df  |-  ( ph  ->  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) ) )

Proof of Theorem wl-mo2df
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 mo2v 2316 . 2  |-  ( E* x ps  <->  E. u A. x ( ps  ->  x  =  u ) )
2 wl-mo2df.2 . . 3  |-  F/ y
ph
3 wl-mo2df.1 . . . 4  |-  F/ x ph
4 wl-mo2df.4 . . . . 5  |-  ( ph  ->  F/ y ps )
5 wl-mo2df.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, 8nfimd 2010 . . . 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 ) )
1514imbi2d 322 . . . . . . 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*wmo 2310
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  df-mo 2314
This theorem is referenced by:  wl-mo2tf  31944
  Copyright terms: Public domain W3C validator