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Theorem mo2OLD 2336
Description: Obsolete proof of mo2 2287 as of 28-May-2019. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
mo2OLD.1  |-  F/ y
ph
Assertion
Ref Expression
mo2OLD  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo2OLD
StepHypRef Expression
1 df-mo 2280 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
2 alnex 1598 . . . . 5  |-  ( A. x  -.  ph  <->  -.  E. x ph )
3 pm2.21 108 . . . . . . 7  |-  ( -. 
ph  ->  ( ph  ->  x  =  y ) )
43alimi 1614 . . . . . 6  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  x  =  y ) )
5 19.8a 1806 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
64, 5syl 16 . . . . 5  |-  ( A. x  -.  ph  ->  E. y A. x ( ph  ->  x  =  y ) )
72, 6sylbir 213 . . . 4  |-  ( -. 
E. x ph  ->  E. y A. x (
ph  ->  x  =  y ) )
8 mo2OLD.1 . . . . 5  |-  F/ y
ph
98eumo0OLD 2313 . . . 4  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
107, 9ja 161 . . 3  |-  ( ( E. x ph  ->  E! x ph )  ->  E. y A. x (
ph  ->  x  =  y ) )
118eu3OLD 2330 . . . 4  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
1211simplbi2com 627 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ph  ->  E! x ph ) )
1310, 12impbii 188 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  E. y A. x ( ph  ->  x  =  y ) )
141, 13bitri 249 1  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1377   E.wex 1596   F/wnf 1599   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-tru 1382  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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