Theorem rmoan 3406

Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmoan	|- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )

Proof of Theorem rmoan

Step	Hyp	Ref	Expression
1		moan 2524	. . 3 |- ( E* x ( x e. A /\ ph ) -> E* x ( ps /\ ( x e. A /\ ph ) ) )
2		an12 838	. . . 4 |- ( ( ps /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( ps /\ ph ) ) )
3	2	mobii 2493	. . 3 |- ( E* x ( ps /\ ( x e. A /\ ph ) ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
4	1, 3	sylib 208	. 2 |- ( E* x ( x e. A /\ ph ) -> E* x ( x e. A /\ ( ps /\ ph ) ) )
5		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6		df-rmo 2920	. 2 |- ( E* x e. A ( ps /\ ph ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
7	4, 5, 6	3imtr4i 281	1 |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 384   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-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-rmo 2920

This theorem is referenced by:  reuxfr2d  4891  reuxfr3d  29329