MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmo5 Unicode version

Theorem rmo5 2756
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 2148 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  ( E. x ( x  e.  A  /\  ph )  ->  E! x ( x  e.  A  /\  ph ) ) )
2 df-rmo 2551 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rex 2549 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2550 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
53, 4imbi12i 316 . 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 268 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 176    /\ wa 358   E.wex 1528    e. wcel 1684   E!weu 2143   E*wmo 2144   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  nrexrmo  2757  cbvrmo  2763  2reurex  27959
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 177  df-mo 2148  df-rex 2549  df-reu 2550  df-rmo 2551
  Copyright terms: Public domain W3C validator