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Theorem rmoim 3134
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 2711 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 imdistan 672 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
32albii 1576 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
41, 3bitri 242 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
5 moim 2328 . . 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 2714 . . 3  |-  ( E* x  e.  A ps  <->  E* x ( x  e.  A  /\  ps )
)
7 df-rmo 2714 . . 3  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
85, 6, 73imtr4g 263 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E* x  e.  A ps  ->  E* x  e.  A ph ) )
94, 8sylbi 189 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 360   A.wal 1550    e. wcel 1726   E*wmo 2283   A.wral 2706   E*wrmo 2709
This theorem is referenced by:  rmoimia  3135  2rmorex  3139  disjss2  4186  catideu  13901  2ndcdisj  17520  evlseu  19938  frlmup4  27231  reuimrmo  27933  2reurex  27936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-ral 2711  df-rmo 2714
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