Theorem euabsn 3462

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
euabsn	|- ( E! x ph <-> E. x { x | ph } = { x } )

Proof of Theorem euabsn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3461	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1461	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2221	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2228	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3409	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2092	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1679	. 2 |- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1, 7	bitr4i 185	1 |- ( E! x ph <-> E. x { x | ph } = { x } )

Syntax hints:   <-> wb 103   = wceq 1284   E.wex 1421   E!weu 1941   {cab 2067   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  eusn  3466  args  4714