Theorem morex 3021

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 2621	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		exancom 1586	. . . 4 |- (E.x(x e. A /\ ph) <-> E.x(ph /\ x e. A))
3	1, 2	bitri 240	. . 3 |- (E.x e. A ph <-> E.x(ph /\ x e. A))
4		nfmo1 2215	. . . . . 6 |- F/xE*xph
5		nfe1 1732	. . . . . 6 |- F/xE.x(ph /\ x e. A)
6	4, 5	nfan 1824	. . . . 5 |- F/x(E*xph /\ E.x(ph /\ x e. A))
7		mopick 2266	. . . . 5 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ph -> x e. A))
8	6, 7	alrimi 1765	. . . 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 2413	. . . . . 6 |- (x = B -> (x e. A <-> B e. A))
12	10, 11	imbi12d 311	. . . . 5 |- (x = B -> ((ph -> x e. A) <-> (ps -> B e. A)))
13	9, 12	spcv 2946	. . . 4 |- (A.x(ph -> x e. A) -> (ps -> B e. A))
14	8, 13	syl 15	. . 3 |- ((E*xph /\ E.x(ph /\ x e. A)) -> (ps -> B e. A))
15	3, 14	sylan2b 461	. 2 |- ((E*xph /\ E.x e. A ph) -> (ps -> B e. A))
16	15	ancoms 439	1 |- ((E.x e. A ph /\ E*xph) -> (ps -> B e. A))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E*wmo 2205  E.wrex 2616  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862

This theorem is referenced by: (None)