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Theorem rmoimi2 2966
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 2189 . . 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 2551 . 2  |-  ( E* x  e.  B ps  <->  E* x ( x  e.  B  /\  ps )
)
5 df-rmo 2551 . 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 1527    e. wcel 1684   E*wmo 2144   E*wrmo 2546
This theorem is referenced by:  disjin  23362
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-rmo 2551
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