Theorem euabsn 3593

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 3592	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1508	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2283	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2291	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3538	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2151	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1729	. 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 1331   E.wex 1468   E!weu 1999   {cab 2125   {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533

This theorem is referenced by:  eusn  3597  args  4908  mapsn  6584