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Theorem rmoan 2976
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 2207 . . 3  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( ps 
/\  ( x  e.  A  /\  ph )
) )
2 an12 772 . . . 4  |-  ( ( ps  /\  ( x  e.  A  /\  ph ) )  <->  ( x  e.  A  /\  ( ps  /\  ph ) ) )
32mobii 2192 . . 3  |-  ( E* x ( ps  /\  ( x  e.  A  /\  ph ) )  <->  E* x
( x  e.  A  /\  ( ps  /\  ph ) ) )
41, 3sylib 188 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  E* x ( x  e.  A  /\  ( ps  /\  ph ) ) )
5 df-rmo 2564 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
6 df-rmo 2564 . 2  |-  ( E* x  e.  A ( ps  /\  ph )  <->  E* x ( x  e.  A  /\  ( ps 
/\  ph ) ) )
74, 5, 63imtr4i 257 1  |-  ( E* x  e.  A ph  ->  E* x  e.  A
( ps  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   E*wmo 2157   E*wrmo 2559
This theorem is referenced by:  reuxfr2d  4573  reuxfr3d  23154
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-rmo 2564
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