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Theorem rmob 3039
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 2450 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
2 simprl 521 . . . 4  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  B  e.  A )
3 eleq1 2227 . . . 4  |-  ( B  =  C  ->  ( B  e.  A  <->  C  e.  A ) )
42, 3syl5ibcom 154 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  ->  C  e.  A ) )
5 simpl 108 . . . 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 525 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  B  e.  A )
8 simpr 109 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  C  e.  A )
9 simpll 519 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  E* x ( x  e.  A  /\  ph )
)
10 simplrr 526 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ps )
11 eleq1 2227 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
12 rmoi.b . . . . . . 7  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 465 . . . . . 6  |-  ( x  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( B  e.  A  /\  ps )
) )
14 eleq1 2227 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
15 rmoi.c . . . . . . 7  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
1614, 15anbi12d 465 . . . . . 6  |-  ( x  =  C  ->  (
( x  e.  A  /\  ph )  <->  ( C  e.  A  /\  ch )
) )
1713, 16mob 2904 . . . . 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 1237 . . . 4  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
1918ex 114 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( C  e.  A  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) ) )
204, 6, 19pm5.21ndd 695 . 2  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
211, 20sylanb 282 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 103    <-> wb 104    = wceq 1342   E*wmo 2014    e. wcel 2135   E*wrmo 2445
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rmo 2450  df-v 2724
This theorem is referenced by:  rmoi  3040
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