Theorem rmoan 2973

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 2123	. . 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 2091	. . 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 2492	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6		df-rmo 2492	. 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 2055   e. wcel 2176   E*wrmo 2487

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-rmo 2492

This theorem is referenced by: (None)