Theorem euabsn 3745

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 3744	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1577	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2377	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2385	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3684	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2243	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1804	. 2 |- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1, 7	bitr4i 187	1 |- ( E! x ph <-> E. x { x | ph } = { x } )

Syntax hints:   <-> wb 105   = wceq 1398   E.wex 1541   E!weu 2079   {cab 2217   {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  eusn  3749  args  5112  mapsn  6902