Theorem euabsn 3646

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 3645	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1516	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2310	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2318	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3587	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2177	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1744	. 2 |- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1, 7	bitr4i 186	1 |- ( E! x ph <-> E. x { x | ph } = { x } )

Syntax hints:   <-> wb 104   = wceq 1343   E.wex 1480   E!weu 2014   {cab 2151   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by:  eusn  3650  args  4973  mapsn  6656