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Theorem rmoan 2762
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 1985 . . 3  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( ps 
/\  ( x  e.  A  /\  ph )
) )
2 an12 503 . . . 4  |-  ( ( ps  /\  ( x  e.  A  /\  ph ) )  <->  ( x  e.  A  /\  ( ps  /\  ph ) ) )
32mobii 1953 . . 3  |-  ( E* x ( ps  /\  ( x  e.  A  /\  ph ) )  <->  E* x
( x  e.  A  /\  ( ps  /\  ph ) ) )
41, 3sylib 131 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( x  e.  A  /\  ( ps  /\  ph ) ) )
5 df-rmo 2331 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
6 df-rmo 2331 . 2  |-  ( E* x  e.  A  ( ps  /\  ph )  <->  E* x ( x  e.  A  /\  ( ps 
/\  ph ) ) )
74, 5, 63imtr4i 194 1  |-  ( E* x  e.  A  ph  ->  E* x  e.  A  ( ps  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    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
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-rmo 2331
This theorem is referenced by: (None)
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