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Theorem rmo5 2926
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 2288 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  ( E. x ( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph ) ) )
2 df-rmo 2715 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rex 2713 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2714 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
53, 4imbi12i 318 . 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 270 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    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1726   E!weu 2283   E*wmo 2284   E.wrex 2708   E!wreu 2709   E*wrmo 2710
This theorem is referenced by:  nrexrmo  2927  cbvrmo  2933  2reurex  27937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 179  df-mo 2288  df-rex 2713  df-reu 2714  df-rmo 2715
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