Theorem rmoim 3027

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 2533	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		imdistan 448	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
3	2	albii 1523	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
4	1, 3	bitri 184	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		moim 2151	. . 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 2536	. . 3 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo 2536	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
8	5, 6, 7	3imtr4g 205	. 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 121	1 |- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   E*wmo 2087   e. wcel 2209   A.wral 2528   E*wrmo 2531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-ral 2533  df-rmo 2536

This theorem is referenced by:  rmoimia  3028  disjss2  4104  rinvmod  14090