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Theorem rmoim 3296
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 2812 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 imdistan 689 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
32albii 1615 . . 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 2334 . . 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 2815 . . 3  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
7 df-rmo 2815 . . 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 369   A.wal 1372    e. wcel 1762   E*wmo 2269   A.wral 2807   E*wrmo 2810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-12 1798
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595  df-eu 2272  df-mo 2273  df-ral 2812  df-rmo 2815
This theorem is referenced by:  rmoimia  3297  2rmorex  3301  disjss2  4413  catideu  14919  evlseu  17949  frlmup4  18595  2ndcdisj  19716  reuimrmo  31605  2reurex  31608
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