Theorem moabex 4651

Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)

Assertion

Ref	Expression
moabex	|- ( E* x ph -> { x | ph } e. _V )

Proof of Theorem moabex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2267	. 2 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		abss 3521	. . . . 5 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x e. { y } ) )
3		elsn 3991	. . . . . . 7 |- ( x e. { y } <-> x = y )
4	3	imbi2i 312	. . . . . 6 |- ( ( ph -> x e. { y } ) <-> ( ph -> x = y ) )
5	4	albii 1611	. . . . 5 |- ( A. x ( ph -> x e. { y } ) <-> A. x ( ph -> x = y ) )
6	2, 5	bitri 249	. . . 4 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x = y ) )
7		snex 4633	. . . . 5 |- { y } e. _V
8	7	ssex 4536	. . . 4 |- ( { x | ph } C_ { y } -> { x | ph } e. _V )
9	6, 8	sylbir 213	. . 3 |- ( A. x ( ph -> x = y ) -> { x | ph } e. _V )
10	9	exlimiv 1689	. 2 |- ( E. y A. x ( ph -> x = y ) -> { x | ph } e. _V )
11	1, 10	sylbi 195	1 |- ( E* x ph -> { x | ph } e. _V )

Syntax hints:   -> wi 4   A.wal 1368   E.wex 1587   e. wcel 1758   E*wmo 2261   {cab 2436   _Vcvv 3070   C_ wss 3428   {csn 3977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980

This theorem is referenced by:  rmorabex  4652  euabex  4653