Theorem rmoan 2816

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 2018	. . 3 |- ( E* x ( x e. A /\ ph ) -> E* x ( ps /\ ( x e. A /\ ph ) ) )
2		an12 529	. . . 4 |- ( ( ps /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( ps /\ ph ) ) )
3	2	mobii 1986	. . 3 |- ( E* x ( ps /\ ( x e. A /\ ph ) ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
4	1, 3	sylib 121	. 2 |- ( E* x ( x e. A /\ ph ) -> E* x ( x e. A /\ ( ps /\ ph ) ) )
5		df-rmo 2368	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
6		df-rmo 2368	. 2 |- ( E* x e. A ( ps /\ ph ) <-> E* x ( x e. A /\ ( ps /\ ph ) ) )
7	4, 5, 6	3imtr4i 200	1 |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1439   E*wmo 1950   E*wrmo 2363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-rmo 2368

This theorem is referenced by: (None)