Theorem moabex 4632

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 2307	. 2 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		abss 3466	. . . . 5 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x e. { y } ) )
3		elsn 3950	. . . . . . 7 |- ( x e. { y } <-> x = y )
4	3	imbi2i 318	. . . . . 6 |- ( ( ph -> x e. { y } ) <-> ( ph -> x = y ) )
5	4	albii 1695	. . . . 5 |- ( A. x ( ph -> x e. { y } ) <-> A. x ( ph -> x = y ) )
6	2, 5	bitri 257	. . . 4 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x = y ) )
7		snex 4614	. . . . 5 |- { y } e. _V
8	7	ssex 4519	. . . 4 |- ( { x | ph } C_ { y } -> { x | ph } e. _V )
9	6, 8	sylbir 218	. . 3 |- ( A. x ( ph -> x = y ) -> { x | ph } e. _V )
10	9	exlimiv 1780	. 2 |- ( E. y A. x ( ph -> x = y ) -> { x | ph } e. _V )
11	1, 10	sylbi 200	1 |- ( E* x ph -> { x | ph } e. _V )

Syntax hints:   -> wi 4   A.wal 1446   E.wex 1667   e. wcel 1891   E*wmo 2301   {cab 2438   _Vcvv 3013   C_ wss 3372   {csn 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-sn 3937  df-pr 3939

This theorem is referenced by:  rmorabex  4633  euabex  4634