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Theorem rmoim 2964
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 2548 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 imdistan 670 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
32albii 1553 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
41, 3bitri 240 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
5 moim 2189 . . 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 2551 . . 3  |-  ( E* x  e.  A ps  <->  E* x ( x  e.  A  /\  ps )
)
7 df-rmo 2551 . . 3  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
85, 6, 73imtr4g 261 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E* x  e.  A ps  ->  E* x  e.  A ph ) )
94, 8sylbi 187 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 358   A.wal 1527    e. wcel 1684   E*wmo 2144   A.wral 2543   E*wrmo 2546
This theorem is referenced by:  rmoimia  2965  2rmorex  2969  disjss2  3996  catideu  13577  2ndcdisj  17182  evlseu  19400  frlmup4  27253  reuimrmo  27956  2reurex  27959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-ral 2548  df-rmo 2551
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