Theorem morex 3390

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 2918	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		exancom 1787	. . . 4 |- ( E. x ( x e. A /\ ph ) <-> E. x ( ph /\ x e. A ) )
3	1, 2	bitri 264	. . 3 |- ( E. x e. A ph <-> E. x ( ph /\ x e. A ) )
4		nfmo1 2481	. . . . . 6 |- F/ x E* x ph
5		nfe1 2027	. . . . . 6 |- F/ x E. x ( ph /\ x e. A )
6	4, 5	nfan 1828	. . . . 5 |- F/ x ( E* x ph /\ E. x ( ph /\ x e. A ) )
7		mopick 2535	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ph -> x e. A ) )
8	6, 7	alrimi 2082	. . . 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 2689	. . . . . 6 |- ( x = B -> ( x e. A <-> B e. A ) )
12	10, 11	imbi12d 334	. . . . 5 |- ( x = B -> ( ( ph -> x e. A ) <-> ( ps -> B e. A ) ) )
13	9, 12	spcv 3299	. . . 4 |- ( A. x ( ph -> x e. A ) -> ( ps -> B e. A ) )
14	8, 13	syl 17	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ x e. A ) ) -> ( ps -> B e. A ) )
15	3, 14	sylan2b 492	. 2 |- ( ( E* x ph /\ E. x e. A ph ) -> ( ps -> B e. A ) )
16	15	ancoms 469	1 |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E*wmo 2471   E.wrex 2913   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202

This theorem is referenced by: (None)