Theorem morex 2776

Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
morex.1	|- B e. _V
morex.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
morex	|- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Distinct variable groups:   x, B   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem morex

Step	Hyp	Ref	Expression
1		df-rex 2354	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		exancom 1539	. . . 4 |- ( E. x ( x e. A /\ ph ) <-> E. x ( ph /\ x e. A ) )
3	1, 2	bitri 182	. . 3 |- ( E. x e. A ph <-> E. x ( ph /\ x e. A ) )
4		nfmo1 1953	. . . . . 6 |- F/ x E* x ph
5		nfe1 1425	. . . . . 6 |- F/ x E. x ( ph /\ x e. A )
6	4, 5	nfan 1497	. . . . 5 |- F/ x ( E* x ph /\ E. x ( ph /\ x e. A ) )
7		mopick 2019	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ph -> x e. A ) )
8	6, 7	alrimi 1455	. . . 4 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> A. x ( ph -> x e. A ) )
9		morex.1	. . . . 5 |- B e. _V
10		morex.2	. . . . . 6 |- ( x = B -> ( ph <-> ps ) )
11		eleq1 2141	. . . . . 6 |- ( x = B -> ( x e. A <-> B e. A ) )
12	10, 11	imbi12d 232	. . . . 5 |- ( x = B -> ( ( ph -> x e. A ) <-> ( ps -> B e. A ) ) )
13	9, 12	spcv 2691	. . . 4 |- ( A. x ( ph -> x e. A ) -> ( ps -> B e. A ) )
14	8, 13	syl 14	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ps -> B e. A ) )
15	3, 14	sylan2b 281	. 2 |- ( ( E* x ph /\ E. x e. A ph ) -> ( ps -> B e. A ) )
16	15	ancoms 264	1 |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E*wmo 1942   E.wrex 2349   _Vcvv 2601

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  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603

This theorem is referenced by: (None)