Theorem morex 2910

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 2450	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		exancom 1596	. . . 4 |- ( E. x ( x e. A /\ ph ) <-> E. x ( ph /\ x e. A ) )
3	1, 2	bitri 183	. . 3 |- ( E. x e. A ph <-> E. x ( ph /\ x e. A ) )
4		nfmo1 2026	. . . . . 6 |- F/ x E* x ph
5		nfe1 1484	. . . . . 6 |- F/ x E. x ( ph /\ x e. A )
6	4, 5	nfan 1553	. . . . 5 |- F/ x ( E* x ph /\ E. x ( ph /\ x e. A ) )
7		mopick 2092	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ph -> x e. A ) )
8	6, 7	alrimi 1510	. . . 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 2229	. . . . . 6 |- ( x = B -> ( x e. A <-> B e. A ) )
12	10, 11	imbi12d 233	. . . . 5 |- ( x = B -> ( ( ph -> x e. A ) <-> ( ps -> B e. A ) ) )
13	9, 12	spcv 2820	. . . 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 285	. 2 |- ( ( E* x ph /\ E. x e. A ph ) -> ( ps -> B e. A ) )
16	15	ancoms 266	1 |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   E*wmo 2015   e. wcel 2136   E.wrex 2445   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728

This theorem is referenced by: (None)