This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rmoim

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017)

Ref Expression
Assertion rmoim 
|- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )

Proof

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