Theorem rmoimi2 2942

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

Hypothesis

Ref	Expression
rmoimi2.1	|- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )

Assertion

Ref	Expression
rmoimi2	|- ( E* x e. B ps -> E* x e. A ph )

Proof of Theorem rmoimi2

Step	Hyp	Ref	Expression
1		rmoimi2.1	. . 3 |- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )
2		moim 2090	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) -> ( E* x ( x e. B /\ ps ) -> E* x ( x e. A /\ ph ) ) )
3	1, 2	ax-mp 5	. 2 |- ( E* x ( x e. B /\ ps ) -> E* x ( x e. A /\ ph ) )
4		df-rmo 2463	. 2 |- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
5		df-rmo 2463	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6	3, 4, 5	3imtr4i 201	1 |- ( E* x e. B ps -> E* x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   E*wmo 2027   e. wcel 2148   E*wrmo 2458

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-rmo 2463

This theorem is referenced by: (None)