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Theorem rmoi 2908
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
rmoi.c  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
rmoi  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )  /\  ( C  e.  A  /\  ch ) )  ->  B  =  C )
Distinct variable groups:    x, A    x, B    x, C    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem rmoi
StepHypRef Expression
1 rmoi.b . . 3  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
2 rmoi.c . . 3  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
31, 2rmob 2907 . 2  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
43biimp3ar 1278 1  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )  /\  ( C  e.  A  /\  ch ) )  ->  B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   E*wrmo 2352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rmo 2357  df-v 2604
This theorem is referenced by: (None)
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