Theorem rmo2ilem 2903

Description: Condition implying restricted "at most one." (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 259	. . . . 5 |- ( ( ( x e. A /\ ph ) -> x = y ) <-> ( x e. A -> ( ph -> x = y ) ) )
2	1	albii 1399	. . . 4 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
3		df-ral 2353	. . . 4 |- ( A. x e. A ( ph -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
4	2, 3	bitr4i 185	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x e. A ( ph -> x = y ) )
5	4	exbii 1536	. 2 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) <-> E. y A. x e. A ( ph -> x = y ) )
6		nfv 1461	. . . . 5 |- F/ y x e. A
7		rmo2.1	. . . . 5 |- F/ y ph
8	6, 7	nfan 1497	. . . 4 |- F/ y ( x e. A /\ ph )
9	8	mo2r 1993	. . 3 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) -> E* x ( x e. A /\ ph ) )
10		df-rmo 2356	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
11	9, 10	sylibr 132	. 2 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) -> E* x e. A ph )
12	5, 11	sylbir 133	1 |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   E*wmo 1942   A.wral 2348   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-ral 2353  df-rmo 2356

This theorem is referenced by:  rmo2i  2904