Theorem rmoan 2952

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 2107	. . 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 2075	. . 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 2476	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6		df-rmo 2476	. 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 2039   e. wcel 2160   E*wrmo 2471

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-rmo 2476

This theorem is referenced by: (None)