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Theorem rmob 2878
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
rmoi.c  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
rmob  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
Distinct variable groups:    x, A    x, B    x, C    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem rmob
StepHypRef Expression
1 df-rmo 2331 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
2 simprl 491 . . . 4  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  B  e.  A )
3 eleq1 2116 . . . 4  |-  ( B  =  C  ->  ( B  e.  A  <->  C  e.  A ) )
42, 3syl5ibcom 148 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  ->  C  e.  A ) )
5 simpl 106 . . . 4  |-  ( ( C  e.  A  /\  ch )  ->  C  e.  A )
65a1i 9 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( ( C  e.  A  /\  ch )  ->  C  e.  A ) )
7 simplrl 495 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  B  e.  A )
8 simpr 107 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  C  e.  A )
9 simpll 489 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  E* x ( x  e.  A  /\  ph )
)
10 simplrr 496 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ps )
11 eleq1 2116 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
12 rmoi.b . . . . . . 7  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 450 . . . . . 6  |-  ( x  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( B  e.  A  /\  ps )
) )
14 eleq1 2116 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
15 rmoi.c . . . . . . 7  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
1614, 15anbi12d 450 . . . . . 6  |-  ( x  =  C  ->  (
( x  e.  A  /\  ph )  <->  ( C  e.  A  /\  ch )
) )
1713, 16mob 2746 . . . . 5  |-  ( ( ( B  e.  A  /\  C  e.  A
)  /\  E* x
( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
187, 8, 9, 7, 10, 17syl212anc 1156 . . . 4  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
1918ex 112 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( C  e.  A  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) ) )
204, 6, 19pm5.21ndd 631 . 2  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
211, 20sylanb 272 1  |-  ( ( E* x  e.  A  ph 
/\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   E*wmo 1917   E*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-v 2576
This theorem is referenced by:  rmoi  2879
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