Theorem morex 15662

Description: Derive membership from uniqueness.

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

Proof of Theorem morex

Step	Hyp	Ref	Expression
1		hbmo1 1802	. . . . . 6 |- (E*xph -> A.xE*xph)
2		hbe1 1363	. . . . . 6 |- (E.x(ph /\ x e. A) -> A.xE.x(ph /\ x e. A))
3	1, 2	hban 1356	. . . . 5 |- ((E*xph /\ E.x(ph /\ x e. A)) -> A.x(E*xph /\ E.x(ph /\ x e. A)))
4		mopick 1833	. . . . 5 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ph -> x e. A))
5	3, 4	19.21ai 1345	. . . 4 |- ((E*xph /\ E.x(ph /\ x e. A)) -> A.x(ph -> x e. A))
6		morex.1	. . . . 5 |- B e. _V
7		morex.2	. . . . . 6 |- (x = B -> (ph <-> ps))
8		eleq1 1957	. . . . . 6 |- (x = B -> (x e. A <-> B e. A))
9	7, 8	imbi12d 688	. . . . 5 |- (x = B -> ((ph -> x e. A) <-> (ps -> B e. A)))
10	6, 9	cla4v 2370	. . . 4 |- (A.x(ph -> x e. A) -> (ps -> B e. A))
11	5, 10	syl 12	. . 3 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ps -> B e. A))
12		df-rex 2110	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
13		exancom 1401	. . . 4 |- (E.x(x e. A /\ ph) <-> E.x(ph /\ x e. A))
14	12, 13	bitri 190	. . 3 |- (E.x e. A ph <-> E.x(ph /\ x e. A))
15	11, 14	sylan2b 501	. 2 |- ((E*xph /\ E.x e. A ph) -> (ps -> B e. A))
16	15	ancoms 484	1 |- ((E.x e. A ph /\ E*xph) -> (ps -> B e. A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E*wmo 1772  E.wrex 2106  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294

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