Theorem morex 2719

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*xph) -> (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 2306	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		exancom 1496	. . . 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 1909	. . . . . 6 |- F/xE*xph
5		nfe1 1382	. . . . . 6 |- F/xE.x(ph /\ x e. A)
6	4, 5	nfan 1454	. . . . 5 |- F/x(E*xph /\ E.x(ph /\ x e. A))
7		mopick 1975	. . . . 5 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ph -> x e. A))
8	6, 7	alrimi 1412	. . . 4 |- ((E*xph /\ 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 2097	. . . . . 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 2640	. . . 4 |- (A.x(ph -> x e. A) -> (ps -> B e. A))
14	8, 13	syl 14	. . 3 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ps -> B e. A))
15	3, 14	sylan2b 271	. 2 |- ((E*xph /\ E.x e. A ph) -> (ps -> B e. A))
16	15	ancoms 255	1 |- ((E.x e. A ph /\ E*xph) -> (ps -> B e. A))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390  E*wmo 1898  E.wrex 2301   _Vcvv 2551

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553

This theorem is referenced by: (None)