Theorem moabex 4927

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 2477	. 2 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		abss 3671	. . . . 5 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x e. { y } ) )
3		velsn 4193	. . . . . . 7 |- ( x e. { y } <-> x = y )
4	3	imbi2i 326	. . . . . 6 |- ( ( ph -> x e. { y } ) <-> ( ph -> x = y ) )
5	4	albii 1747	. . . . 5 |- ( A. x ( ph -> x e. { y } ) <-> A. x ( ph -> x = y ) )
6	2, 5	bitri 264	. . . 4 |- ( { x | ph } C_ { y } <-> A. x ( ph -> x = y ) )
7		snex 4908	. . . . 5 |- { y } e. _V
8	7	ssex 4802	. . . 4 |- ( { x | ph } C_ { y } -> { x | ph } e. _V )
9	6, 8	sylbir 225	. . 3 |- ( A. x ( ph -> x = y ) -> { x | ph } e. _V )
10	9	exlimiv 1858	. 2 |- ( E. y A. x ( ph -> x = y ) -> { x | ph } e. _V )
11	1, 10	sylbi 207	1 |- ( E* x ph -> { x | ph } e. _V )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   e. wcel 1990   E*wmo 2471   {cab 2608   _Vcvv 3200   C_ wss 3574   {csn 4177

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  rmorabex  4928  euabex  4929