Theorem euabsn 3518

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 3517	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1467	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2231	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2239	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3463	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2100	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1687	. 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 1290   E.wex 1427   E!weu 1949   {cab 2075   {csn 3452

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-sn 3458

This theorem is referenced by:  eusn  3522  args  4816  mapsn  6463