Theorem rmo2ilem 3119

Description: Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)

Hypothesis

Ref	Expression
rmo2.1	|- F/ y ph

Assertion

Ref	Expression
rmo2ilem	|- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem rmo2ilem

Step	Hyp	Ref	Expression
1		impexp 263	. . . . 5 |- ( ( ( x e. A /\ ph ) -> x = y ) <-> ( x e. A -> ( ph -> x = y ) ) )
2	1	albii 1516	. . . 4 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
3		df-ral 2513	. . . 4 |- ( A. x e. A ( ph -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
4	2, 3	bitr4i 187	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x e. A ( ph -> x = y ) )
5	4	exbii 1651	. 2 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) <-> E. y A. x e. A ( ph -> x = y ) )
6		nfv 1574	. . . . 5 |- F/ y x e. A
7		rmo2.1	. . . . 5 |- F/ y ph
8	6, 7	nfan 1611	. . . 4 |- F/ y ( x e. A /\ ph )
9	8	mo2r 2130	. . 3 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) -> E* x ( x e. A /\ ph ) )
10		df-rmo 2516	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
11	9, 10	sylibr 134	. 2 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) -> E* x e. A ph )
12	5, 11	sylbir 135	1 |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   F/wnf 1506   E.wex 1538   E*wmo 2078   e. wcel 2200   A.wral 2508   E*wrmo 2511

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-ral 2513  df-rmo 2516

This theorem is referenced by:  rmo2i  3120