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Theorem rmo5 2574
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )

Proof of Theorem rmo5
StepHypRef Expression
1 df-mo 1947 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  ( E. x ( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph ) ) )
2 df-rmo 2361 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rex 2359 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2360 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
53, 4imbi12i 237 . 2  |-  ( ( E. x  e.  A  ph 
->  E! x  e.  A  ph )  <->  ( E. x
( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph )
) )
61, 2, 53bitr4i 210 1  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   E!weu 1943   E*wmo 1944   E.wrex 2354   E!wreu 2355   E*wrmo 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-mo 1947  df-rex 2359  df-reu 2360  df-rmo 2361
This theorem is referenced by:  nrexrmo  2575  cbvrmo  2581  bdrmo  10914
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