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Theorem rmoan 2816
Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmoan  |-  ( E* x  e.  A  ph  ->  E* x  e.  A  ( ps  /\  ph )
)

Proof of Theorem rmoan
StepHypRef Expression
1 moan 2018 . . 3  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( ps 
/\  ( x  e.  A  /\  ph )
) )
2 an12 529 . . . 4  |-  ( ( ps  /\  ( x  e.  A  /\  ph ) )  <->  ( x  e.  A  /\  ( ps  /\  ph ) ) )
32mobii 1986 . . 3  |-  ( E* x ( ps  /\  ( x  e.  A  /\  ph ) )  <->  E* x
( x  e.  A  /\  ( ps  /\  ph ) ) )
41, 3sylib 121 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( x  e.  A  /\  ( ps  /\  ph ) ) )
5 df-rmo 2368 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
6 df-rmo 2368 . 2  |-  ( E* x  e.  A  ( ps  /\  ph )  <->  E* x ( x  e.  A  /\  ( ps 
/\  ph ) ) )
74, 5, 63imtr4i 200 1  |-  ( E* x  e.  A  ph  ->  E* x  e.  A  ( ps  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1439   E*wmo 1950   E*wrmo 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-rmo 2368
This theorem is referenced by: (None)
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