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Theorem rmoi 2972
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 2971 . 2  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
43biimp3ar 1307 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E*wrmo 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rmo 2399  df-v 2660
This theorem is referenced by: (None)
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