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Theorem rmoi 3335
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 3334 . 2  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
43biimp3ar 1365 1  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )  /\  ( C  e.  A  /\  ch ) )  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E*wrmo 2717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rmo 2722  df-v 3024
This theorem is referenced by:  eqsqrtd  13374  efgred2  17346  0frgp  17372  frgpnabllem2  17453  frgpcyg  19086  cdleme0moN  33703  proot1mul  35986
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