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Theorem rmoimi2 2765
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 1980 . . 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 7 . 2  |-  ( E* x ( x  e.  B  /\  ps )  ->  E* x ( x  e.  A  /\  ph ) )
4 df-rmo 2331 . 2  |-  ( E* x  e.  B  ps  <->  E* x ( x  e.  B  /\  ps )
)
5 df-rmo 2331 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
63, 4, 53imtr4i 194 1  |-  ( E* x  e.  B  ps  ->  E* x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    e. wcel 1409   E*wmo 1917   E*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-rmo 2331
This theorem is referenced by: (None)
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