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Theorem rmo5 3045
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 2268 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  ( E. x ( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph ) ) )
2 df-rmo 2781 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rex 2779 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2780 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
53, 4imbi12i 327 . 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 280 1  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   E.wex 1659    e. wcel 1867   E!weu 2263   E*wmo 2264   E.wrex 2774   E!wreu 2775   E*wrmo 2776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-mo 2268  df-rex 2779  df-reu 2780  df-rmo 2781
This theorem is referenced by:  nrexrmo  3046  cbvrmo  3052  ddemeas  29048  2reurex  38223  iccpartdisj  38371
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