Theorem rmoan

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

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

Proof

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