Theorem rmoim 2931

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 2453	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		imdistan 442	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
3	2	albii 1463	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
4	1, 3	bitri 183	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		moim 2083	. . 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 2456	. . 3 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo 2456	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
8	5, 6, 7	3imtr4g 204	. 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 120	1 |- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E*wmo 2020   e. wcel 2141   A.wral 2448   E*wrmo 2451

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-ral 2453  df-rmo 2456

This theorem is referenced by:  rmoimia  2932  disjss2  3969