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Theorem rmoim 3249
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 2759 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 imdistan 687 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
32albii 1661 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
41, 3bitri 249 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
5 moim 2291 . . 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 2762 . . 3  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
7 df-rmo 2762 . . 3  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
85, 6, 73imtr4g 270 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E* x  e.  A  ps  ->  E* x  e.  A  ph ) )
94, 8sylbi 195 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E* x  e.  A  ps  ->  E* x  e.  A  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403    e. wcel 1842   E*wmo 2239   A.wral 2754   E*wrmo 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243  df-ral 2759  df-rmo 2762
This theorem is referenced by:  rmoimia  3250  2rmorex  3254  disjss2  4369  catideu  15289  evlseu  18505  frlmup4  19128  2ndcdisj  20249  reuimrmo  37551  2reurex  37554
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