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Theorem rmob2 3373
Description: Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
Hypotheses
Ref Expression
rmoi2.1  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
rmoi2.2  |-  ( ph  ->  B  e.  A )
rmoi2.3  |-  ( ph  ->  E* x  e.  A  ps )
rmoi2.4  |-  ( ph  ->  x  e.  A )
rmoi2.5  |-  ( ph  ->  ps )
Assertion
Ref Expression
rmob2  |-  ( ph  ->  ( x  =  B  <->  ch ) )
Distinct variable groups:    x, A    x, B    ch, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rmob2
StepHypRef Expression
1 rmoi2.2 . . 3  |-  ( ph  ->  B  e.  A )
2 rmoi2.3 . . . 4  |-  ( ph  ->  E* x  e.  A  ps )
3 df-rmo 2757 . . . 4  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
42, 3sylib 201 . . 3  |-  ( ph  ->  E* x ( x  e.  A  /\  ps ) )
5 rmoi2.4 . . 3  |-  ( ph  ->  x  e.  A )
6 rmoi2.5 . . 3  |-  ( ph  ->  ps )
7 eleq1 2528 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
8 rmoi2.1 . . . . 5  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
97, 8anbi12d 722 . . . 4  |-  ( x  =  B  ->  (
( x  e.  A  /\  ps )  <->  ( B  e.  A  /\  ch )
) )
109mob2 3230 . . 3  |-  ( ( B  e.  A  /\  E* x ( x  e.  A  /\  ps )  /\  ( x  e.  A  /\  ps ) )  -> 
( x  =  B  <-> 
( B  e.  A  /\  ch ) ) )
111, 4, 5, 6, 10syl112anc 1280 . 2  |-  ( ph  ->  ( x  =  B  <-> 
( B  e.  A  /\  ch ) ) )
121biantrurd 515 . 2  |-  ( ph  ->  ( ch  <->  ( B  e.  A  /\  ch )
) )
1311, 12bitr4d 264 1  |-  ( ph  ->  ( x  =  B  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   E*wmo 2311   E*wrmo 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rmo 2757  df-v 3059
This theorem is referenced by:  rmoi2  3374
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