Theorem rmoim 2978

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 2490	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		imdistan 444	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
3	2	albii 1494	. . 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 2119	. . 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 2493	. . 3 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo 2493	. . 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 1371   E*wmo 2056   e. wcel 2177   A.wral 2485   E*wrmo 2488

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-ral 2490  df-rmo 2493

This theorem is referenced by:  rmoimia  2979  disjss2  4030  rinvmod  13720