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Theorem rmoimi2 2858
Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
rmoimi2.1  |-  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps ) )
Assertion
Ref Expression
rmoimi2  |-  ( E* x  e.  B  ps  ->  E* x  e.  A  ph )

Proof of Theorem rmoimi2
StepHypRef Expression
1 rmoimi2.1 . . 3  |-  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps ) )
2 moim 2039 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps )
)  ->  ( E* x ( x  e.  B  /\  ps )  ->  E* x ( x  e.  A  /\  ph ) ) )
31, 2ax-mp 5 . 2  |-  ( E* x ( x  e.  B  /\  ps )  ->  E* x ( x  e.  A  /\  ph ) )
4 df-rmo 2399 . 2  |-  ( E* x  e.  B  ps  <->  E* x ( x  e.  B  /\  ps )
)
5 df-rmo 2399 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
63, 4, 53imtr4i 200 1  |-  ( E* x  e.  B  ps  ->  E* x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    e. wcel 1463   E*wmo 1976   E*wrmo 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-rmo 2399
This theorem is referenced by: (None)
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