Theorem rmoim 3407

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 2917	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		imdistan 725	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
3	2	albii 1747	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
4	1, 3	bitri 264	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		moim 2519	. . 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 2920	. . 3 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo 2920	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
8	5, 6, 7	3imtr4g 285	. 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 207	1 |- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   A.wral 2912   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rmo 2920

This theorem is referenced by:  rmoimia  3408  2rmorex  3412  disjss2  4623  catideu  16336  evlseu  19516  frlmup4  20140  2ndcdisj  21259  poimirlem18  33427  poimirlem21  33430  reuimrmo  41178  2reurex  41181