Theorem rmoim 2791

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

Step	Hyp	Ref	Expression
1		df-ral 2353	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		imdistan 432	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
3	2	albii 1399	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
4	1, 3	bitri 182	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		moim 2005	. . 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 2356	. . 3 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo 2356	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
8	5, 6, 7	3imtr4g 203	. 2 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) -> ( E* x e. A ps -> E* x e. A ph ) )
9	4, 8	sylbi 119	1 |- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   e. wcel 1433   E*wmo 1942   A.wral 2348   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-ral 2353  df-rmo 2356

This theorem is referenced by:  rmoimia  2792  disjss2  3769