Theorem morex 2725

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 2312	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		exancom 1499	. . . 4 |- ( E. x ( x e. A /\ ph ) <-> E. x ( ph /\ x e. A ) )
3	1, 2	bitri 173	. . 3 |- ( E. x e. A ph <-> E. x ( ph /\ x e. A ) )
4		nfmo1 1912	. . . . . 6 |- F/ x E* x ph
5		nfe1 1385	. . . . . 6 |- F/ x E. x ( ph /\ x e. A )
6	4, 5	nfan 1457	. . . . 5 |- F/ x ( E* x ph /\ E. x ( ph /\ x e. A ) )
7		mopick 1978	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ph -> x e. A ) )
8	6, 7	alrimi 1415	. . . 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 2100	. . . . . 6 |- ( x = B -> ( x e. A <-> B e. A ) )
12	10, 11	imbi12d 223	. . . . 5 |- ( x = B -> ( ( ph -> x e. A ) <-> ( ps -> B e. A ) ) )
13	9, 12	spcv 2646	. . . 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 271	. 2 |- ( ( E* x ph /\ E. x e. A ph ) -> ( ps -> B e. A ) )
16	15	ancoms 255	1 |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E*wmo 1901   E.wrex 2307   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by: (None)