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Theorem rmo2ilem 3038
Description: Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)
Hypothesis
Ref Expression
rmo2.1  |-  F/ y
ph
Assertion
Ref Expression
rmo2ilem  |-  ( E. y A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem rmo2ilem
StepHypRef Expression
1 impexp 261 . . . . 5  |-  ( ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
21albii 1457 . . . 4  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
3 df-ral 2447 . . . 4  |-  ( A. x  e.  A  ( ph  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
42, 3bitr4i 186 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x  e.  A  ( ph  ->  x  =  y ) )
54exbii 1592 . 2  |-  ( E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
6 nfv 1515 . . . . 5  |-  F/ y  x  e.  A
7 rmo2.1 . . . . 5  |-  F/ y
ph
86, 7nfan 1552 . . . 4  |-  F/ y ( x  e.  A  /\  ph )
98mo2r 2065 . . 3  |-  ( E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  ->  E* x ( x  e.  A  /\  ph )
)
10 df-rmo 2450 . . 3  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
119, 10sylibr 133 . 2  |-  ( E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  ->  E* x  e.  A  ph )
125, 11sylbir 134 1  |-  ( E. y A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1340    = wceq 1342   F/wnf 1447   E.wex 1479   E*wmo 2014    e. wcel 2135   A.wral 2442   E*wrmo 2445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-ral 2447  df-rmo 2450
This theorem is referenced by:  rmo2i  3039
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