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Theorem wl-mo2dnae 30184
Description: Version of mo2 2229 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-mo2dnae.1  |-  F/ x ph
wl-mo2dnae.2  |-  F/ y
ph
wl-mo2dnae.3  |-  ( ph  ->  -.  A. x  x  =  y )
wl-mo2dnae.4  |-  ( ph  ->  F/ y ps )
Assertion
Ref Expression
wl-mo2dnae  |-  ( ph  ->  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) ) )

Proof of Theorem wl-mo2dnae
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 mo2v 2225 . 2  |-  ( E* x ps  <->  E. u A. x ( ps  ->  x  =  u ) )
2 wl-mo2dnae.2 . . 3  |-  F/ y
ph
3 wl-mo2dnae.1 . . . 4  |-  F/ x ph
4 wl-mo2dnae.4 . . . . 5  |-  ( ph  ->  F/ y ps )
5 wl-mo2dnae.3 . . . . . 6  |-  ( ph  ->  -.  A. x  x  =  y )
6 nfeqf1 2049 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y  x  =  u )
76naecoms 2059 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ y  x  =  u )
85, 7syl 16 . . . . 5  |-  ( ph  ->  F/ y  x  =  u )
94, 8nfimd 1925 . . . 4  |-  ( ph  ->  F/ y ( ps 
->  x  =  u
) )
103, 9nfald 1959 . . 3  |-  ( ph  ->  F/ y A. x
( ps  ->  x  =  u ) )
11 nfnae 2064 . . . . . . 7  |-  F/ x  -.  A. x  x  =  y
12 nfeqf2 2047 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/ x  u  =  y )
1311, 12nfan1 1935 . . . . . 6  |-  F/ x
( -.  A. x  x  =  y  /\  u  =  y )
14 equequ2 1807 . . . . . . . 8  |-  ( u  =  y  ->  (
x  =  u  <->  x  =  y ) )
1514imbi2d 314 . . . . . . 7  |-  ( u  =  y  ->  (
( ps  ->  x  =  u )  <->  ( ps  ->  x  =  y ) ) )
1615adantl 464 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  u  =  y )  -> 
( ( ps  ->  x  =  u )  <->  ( ps  ->  x  =  y ) ) )
1713, 16albid 1893 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  u  =  y )  -> 
( A. x ( ps  ->  x  =  u )  <->  A. x
( ps  ->  x  =  y ) ) )
185, 17sylan 469 . . . 4  |-  ( (
ph  /\  u  =  y )  ->  ( A. x ( ps  ->  x  =  u )  <->  A. x
( ps  ->  x  =  y ) ) )
1918ex 432 . . 3  |-  ( ph  ->  ( u  =  y  ->  ( A. x
( ps  ->  x  =  u )  <->  A. x
( ps  ->  x  =  y ) ) ) )
202, 10, 19cbvexd 2033 . 2  |-  ( ph  ->  ( E. u A. x ( ps  ->  x  =  u )  <->  E. y A. x ( ps  ->  x  =  y ) ) )
211, 20syl5bb 257 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 184    /\ wa 367   A.wal 1397   E.wex 1620   F/wnf 1624   E*wmo 2219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-eu 2222  df-mo 2223
This theorem is referenced by: (None)
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