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Theorem rmoim 2814
Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
rmoim  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E* x  e.  A  ps  ->  E* x  e.  A  ph )
)

Proof of Theorem rmoim
StepHypRef Expression
1 df-ral 2364 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 imdistan 433 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
32albii 1404 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
41, 3bitri 182 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
5 moim 2012 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E* x ( x  e.  A  /\  ps )  ->  E* x ( x  e.  A  /\  ph ) ) )
6 df-rmo 2367 . . 3  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
7 df-rmo 2367 . . 3  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
85, 6, 73imtr4g 203 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E* x  e.  A  ps  ->  E* x  e.  A  ph ) )
94, 8sylbi 119 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E* x  e.  A  ps  ->  E* x  e.  A  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    e. wcel 1438   E*wmo 1949   A.wral 2359   E*wrmo 2362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-ral 2364  df-rmo 2367
This theorem is referenced by:  rmoimia  2815  disjss2  3817
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