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Theorem eu3v 2337
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on  ph. (Revised by Wolf Lammen, 29-May-2019.)
Assertion
Ref Expression
eu3v  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem eu3v
StepHypRef Expression
1 eu5 2335 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
2 mo2v 2316 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
32anbi2i 705 . 2  |-  ( ( E. x ph  /\  E* x ph )  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
41, 3bitri 257 1  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673   E!weu 2309   E*wmo 2310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-eu 2313  df-mo 2314
This theorem is referenced by:  eqeu  3220  reu3  3239  eunex  4609  bj-eunex  31458
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