Theorem rmoan 2938

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 2095	. . 3 |- ( E* x ( x e. A /\ ph ) -> E* x ( ps /\ ( x e. A /\ ph ) ) )
2		an12 561	. . . 4 |- ( ( ps /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( ps /\ ph ) ) )
3	2	mobii 2063	. . 3 |- ( E* x ( ps /\ ( x e. A /\ ph ) ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
4	1, 3	sylib 122	. 2 |- ( E* x ( x e. A /\ ph ) -> E* x ( x e. A /\ ( ps /\ ph ) ) )
5		df-rmo 2463	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6		df-rmo 2463	. 2 |- ( E* x e. A ( ps /\ ph ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
7	4, 5, 6	3imtr4i 201	1 |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 104   E*wmo 2027   e. wcel 2148   E*wrmo 2458

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-rmo 2463

This theorem is referenced by: (None)