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Theorem wl-sb8eut 31813
Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2309. (Contributed by Wolf Lammen, 11-Aug-2019.)
Assertion
Ref Expression
wl-sb8eut  |-  ( A. x F/ y ph  ->  ( E! x ph  <->  E! y [ y  /  x ] ph ) )

Proof of Theorem wl-sb8eut
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfnf1 1958 . . . . . 6  |-  F/ y F/ y ph
21nfal 2007 . . . . 5  |-  F/ y A. x F/ y
ph
3 equsb3 2238 . . . . . . 7  |-  ( [ v  /  x ]
x  =  u  <->  v  =  u )
43sblbis 2209 . . . . . 6  |-  ( [ v  /  x ]
( ph  <->  x  =  u
)  <->  ( [ v  /  x ] ph  <->  v  =  u ) )
5 nfa1 1956 . . . . . . . 8  |-  F/ x A. x F/ y ph
6 sp 1914 . . . . . . . 8  |-  ( A. x F/ y ph  ->  F/ y ph )
75, 6nfsbd 2248 . . . . . . 7  |-  ( A. x F/ y ph  ->  F/ y [ v  /  x ] ph )
8 nfvd 1756 . . . . . . 7  |-  ( A. x F/ y ph  ->  F/ y  v  =  u )
97, 8nfbid 1993 . . . . . 6  |-  ( A. x F/ y ph  ->  F/ y ( [ v  /  x ] ph  <->  v  =  u ) )
104, 9nfxfrd 1691 . . . . 5  |-  ( A. x F/ y ph  ->  F/ y [ v  /  x ] ( ph  <->  x  =  u ) )
11 sbequ 2181 . . . . . 6  |-  ( v  =  y  ->  ( [ v  /  x ] ( ph  <->  x  =  u )  <->  [ y  /  x ] ( ph  <->  x  =  u ) ) )
1211a1i 11 . . . . 5  |-  ( A. x F/ y ph  ->  ( v  =  y  -> 
( [ v  /  x ] ( ph  <->  x  =  u )  <->  [ y  /  x ] ( ph  <->  x  =  u ) ) ) )
132, 10, 12cbvald 2090 . . . 4  |-  ( A. x F/ y ph  ->  ( A. v [ v  /  x ] (
ph 
<->  x  =  u )  <->  A. y [ y  /  x ] ( ph  <->  x  =  u ) ) )
14 nfv 1755 . . . . . 6  |-  F/ v ( ph  <->  x  =  u )
1514sb8 2229 . . . . 5  |-  ( A. x ( ph  <->  x  =  u )  <->  A. v [ v  /  x ] ( ph  <->  x  =  u ) )
1615bicomi 205 . . . 4  |-  ( A. v [ v  /  x ] ( ph  <->  x  =  u )  <->  A. x
( ph  <->  x  =  u
) )
17 equsb3 2238 . . . . . 6  |-  ( [ y  /  x ]
x  =  u  <->  y  =  u )
1817sblbis 2209 . . . . 5  |-  ( [ y  /  x ]
( ph  <->  x  =  u
)  <->  ( [ y  /  x ] ph  <->  y  =  u ) )
1918albii 1685 . . . 4  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  u )  <->  A. y
( [ y  /  x ] ph  <->  y  =  u ) )
2013, 16, 193bitr3g 290 . . 3  |-  ( A. x F/ y ph  ->  ( A. x ( ph  <->  x  =  u )  <->  A. y
( [ y  /  x ] ph  <->  y  =  u ) ) )
2120exbidv 1762 . 2  |-  ( A. x F/ y ph  ->  ( E. u A. x
( ph  <->  x  =  u
)  <->  E. u A. y
( [ y  /  x ] ph  <->  y  =  u ) ) )
22 df-eu 2280 . 2  |-  ( E! x ph  <->  E. u A. x ( ph  <->  x  =  u ) )
23 df-eu 2280 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. u A. y ( [ y  /  x ] ph  <->  y  =  u ) )
2421, 22, 233bitr4g 291 1  |-  ( A. x F/ y ph  ->  ( E! x ph  <->  E! y [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661   [wsb 1790   E!weu 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280
This theorem is referenced by:  wl-sb8mot  31814
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