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Theorem rmoi 3252
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 3251 . 2  |-  ( ( E* x  e.  A ph  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
43biimp3ar 1285 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    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E*wrmo 2710
This theorem is referenced by:  eqsqrd  12176  efgred2  15390  0frgp  15416  frgpnabllem2  15490  frgpcyg  16859  qtophmeo  17854  proot1mul  27506  cdleme0moN  31096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rmo 2715  df-v 2960
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