Theorem rmoimi2 3409

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 2519	. . 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 2920	. 2 |- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
5		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6	3, 4, 5	3imtr4i 281	1 |- ( E* x e. B ps -> E* x e. A ph )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   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-rmo 2920

This theorem is referenced by:  disjin  29399  disjin2  29400