Theorem rmoimi2 2933

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 2083	. . 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 2456	. 2 |- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
5		df-rmo 2456	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6	3, 4, 5	3imtr4i 200	1 |- ( E* x e. B ps -> E* x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E*wmo 2020   e. wcel 2141   E*wrmo 2451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-rmo 2456

This theorem is referenced by: (None)