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Theorem rmoimi2 2979
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 2202 . . 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 8 . 2  |-  ( E* x ( x  e.  B  /\  ps )  ->  E* x ( x  e.  A  /\  ph ) )
4 df-rmo 2564 . 2  |-  ( E* x  e.  B ps  <->  E* x ( x  e.  B  /\  ps )
)
5 df-rmo 2564 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
63, 4, 53imtr4i 257 1  |-  ( E* x  e.  B ps  ->  E* x  e.  A ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    e. wcel 1696   E*wmo 2157   E*wrmo 2559
This theorem is referenced by:  disjin  23377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-rmo 2564
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