Theorem rmoimi2 3006

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 2142	. . 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 2516	. 2 |- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
5		df-rmo 2516	. 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 1393   E*wmo 2078   e. wcel 2200   E*wrmo 2511

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-rmo 2516

This theorem is referenced by: (None)