Theorem morex 3010

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 2534	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		exancom 1661	. . . 4 |- ( E. x ( x e. A /\ ph ) <-> E. x ( ph /\ x e. A ) )
3	1, 2	bitri 184	. . 3 |- ( E. x e. A ph <-> E. x ( ph /\ x e. A ) )
4		nfmo1 2098	. . . . . 6 |- F/ x E* x ph
5		nfe1 1549	. . . . . 6 |- F/ x E. x ( ph /\ x e. A )
6	4, 5	nfan 1618	. . . . 5 |- F/ x ( E* x ph /\ E. x ( ph /\ x e. A ) )
7		mopick 2165	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ph -> x e. A ) )
8	6, 7	alrimi 1575	. . . 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 2301	. . . . . 6 |- ( x = B -> ( x e. A <-> B e. A ) )
12	10, 11	imbi12d 234	. . . . 5 |- ( x = B -> ( ( ph -> x e. A ) <-> ( ps -> B e. A ) ) )
13	9, 12	spcv 2919	. . . 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 287	. 2 |- ( ( E* x ph /\ E. x e. A ph ) -> ( ps -> B e. A ) )
16	15	ancoms 268	1 |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   E*wmo 2087   e. wcel 2209   E.wrex 2529   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823

This theorem is referenced by: (None)