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Theorem wl-mo2t 30185
Description: Closed form of mo2 2229. (Contributed by Wolf Lammen, 18-Aug-2019.)
Assertion
Ref Expression
wl-mo2t  |-  ( A. x F/ y ph  ->  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem wl-mo2t
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 mo2v 2225 . 2  |-  ( E* x ph  <->  E. u A. x ( ph  ->  x  =  u ) )
2 nfnf1 1907 . . . 4  |-  F/ y F/ y ph
32nfal 1955 . . 3  |-  F/ y A. x F/ y
ph
4 nfa1 1905 . . . 4  |-  F/ x A. x F/ y ph
5 sp 1867 . . . . 5  |-  ( A. x F/ y ph  ->  F/ y ph )
6 nfvd 1716 . . . . 5  |-  ( A. x F/ y ph  ->  F/ y  x  =  u )
75, 6nfimd 1925 . . . 4  |-  ( A. x F/ y ph  ->  F/ y ( ph  ->  x  =  u ) )
84, 7nfald 1959 . . 3  |-  ( A. x F/ y ph  ->  F/ y A. x (
ph  ->  x  =  u ) )
9 equequ2 1807 . . . . . 6  |-  ( u  =  y  ->  (
x  =  u  <->  x  =  y ) )
109imbi2d 314 . . . . 5  |-  ( u  =  y  ->  (
( ph  ->  x  =  u )  <->  ( ph  ->  x  =  y ) ) )
1110albidv 1721 . . . 4  |-  ( u  =  y  ->  ( A. x ( ph  ->  x  =  u )  <->  A. x
( ph  ->  x  =  y ) ) )
1211a1i 11 . . 3  |-  ( A. x F/ y ph  ->  ( u  =  y  -> 
( A. x (
ph  ->  x  =  u )  <->  A. x ( ph  ->  x  =  y ) ) ) )
133, 8, 12cbvexd 2033 . 2  |-  ( A. x F/ y ph  ->  ( E. u A. x
( ph  ->  x  =  u )  <->  E. y A. x ( ph  ->  x  =  y ) ) )
141, 13syl5bb 257 1  |-  ( A. x F/ y ph  ->  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   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:  wl-mo3t  30186
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