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Theorem rmob 3092
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 2564 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
2 simprl 732 . . . 4  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  B  e.  A )
3 eleq1 2356 . . . 4  |-  ( B  =  C  ->  ( B  e.  A  <->  C  e.  A ) )
42, 3syl5ibcom 211 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  ->  C  e.  A ) )
5 simpl 443 . . . 4  |-  ( ( C  e.  A  /\  ch )  ->  C  e.  A )
65a1i 10 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( ( C  e.  A  /\  ch )  ->  C  e.  A ) )
7 simplrl 736 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  B  e.  A )
8 simpr 447 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  C  e.  A )
9 simpll 730 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  E* x ( x  e.  A  /\  ph )
)
10 simplrr 737 . . . . 5  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ps )
11 eleq1 2356 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
12 rmoi.b . . . . . . 7  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 691 . . . . . 6  |-  ( x  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( B  e.  A  /\  ps )
) )
14 eleq1 2356 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
15 rmoi.c . . . . . . 7  |-  ( x  =  C  ->  ( ph 
<->  ch ) )
1614, 15anbi12d 691 . . . . . 6  |-  ( x  =  C  ->  (
( x  e.  A  /\  ph )  <->  ( C  e.  A  /\  ch )
) )
1713, 16mob 2960 . . . . 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 1192 . . . 4  |-  ( ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  /\  C  e.  A )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
1918ex 423 . . 3  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( C  e.  A  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) ) )
204, 6, 19pm5.21ndd 343 . 2  |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps )
)  ->  ( B  =  C  <->  ( C  e.  A  /\  ch )
) )
211, 20sylanb 458 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E*wmo 2157   E*wrmo 2559
This theorem is referenced by:  rmoi  3093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rmo 2564  df-v 2803
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