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Theorem wl-sb8mot 29957
Description: Substitution of variable in universal quantifier. Closed form of sb8mo 2317.

This theorem relates to wl-mo3t 29955, since replacing  ph with  [ y  /  x ] ph in the latter yields subexpressions like  [ x  / 
y ] [ y  /  x ] ph, which can be reduced to  ph via sbft 2093 and sbco 2129. So  E* x ph  <->  E* y [ y  /  x ] ph is provable from wl-mo3t 29955 in a simple fashion, unfortunately only if  x and  y are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 29955 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.)

Assertion
Ref Expression
wl-sb8mot  |-  ( A. x F/ y ph  ->  ( E* x ph  <->  E* y [ y  /  x ] ph ) )

Proof of Theorem wl-sb8mot
StepHypRef Expression
1 wl-sb8et 29935 . . 3  |-  ( A. x F/ y ph  ->  ( E. x ph  <->  E. y [ y  /  x ] ph ) )
2 wl-sb8eut 29956 . . 3  |-  ( A. x F/ y ph  ->  ( E! x ph  <->  E! y [ y  /  x ] ph ) )
31, 2imbi12d 320 . 2  |-  ( A. x F/ y ph  ->  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) ) )
4 df-mo 2280 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
5 df-mo 2280 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
63, 4, 53bitr4g 288 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 184   A.wal 1377   E.wex 1596   F/wnf 1599   [wsb 1711   E!weu 2275   E*wmo 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280
This theorem is referenced by: (None)
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