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Theorem rmo5 2623
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 1981 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  ( E. x ( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph ) ) )
2 df-rmo 2401 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rex 2399 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2400 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
53, 4imbi12i 238 . 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 211 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 103    <-> wb 104   E.wex 1453    e. wcel 1465   E!weu 1977   E*wmo 1978   E.wrex 2394   E!wreu 2395   E*wrmo 2396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-mo 1981  df-rex 2399  df-reu 2400  df-rmo 2401
This theorem is referenced by:  nrexrmo  2624  cbvrmo  2630  bdrmo  12981
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