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Theorem sb8euOLD 2320
Description: Obsolete proof of sb8eu 2319 as of 24-Aug-2019. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 8-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8euOLD  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )

Proof of Theorem sb8euOLD
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1708 . . . . 5  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2168 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 equsb3 2177 . . . . . . 7  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
43sblbis 2146 . . . . . 6  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  w  =  z ) )
5 sb8eu.1 . . . . . . . 8  |-  F/ y
ph
65nfsb 2185 . . . . . . 7  |-  F/ y [ w  /  x ] ph
7 nfv 1708 . . . . . . 7  |-  F/ y  w  =  z
86, 7nfbi 1935 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  w  =  z )
94, 8nfxfr 1646 . . . . 5  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
10 nfv 1708 . . . . 5  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
11 sbequ 2118 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
129, 10, 11cbval 2022 . . . 4  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
13 equsb3 2177 . . . . . 6  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1413sblbis 2146 . . . . 5  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1514albii 1641 . . . 4  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
162, 12, 153bitri 271 . . 3  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1716exbii 1668 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
18 df-eu 2287 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
19 df-eu 2287 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
2017, 18, 193bitr4i 277 1  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1393   E.wex 1613   F/wnf 1617   [wsb 1740   E!weu 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287
This theorem is referenced by: (None)
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