Theorem euabsn 3653

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 3652	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1521	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2314	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2322	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3594	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2182	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1749	. 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 1348   E.wex 1485   E!weu 2019   {cab 2156   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by:  eusn  3657  args  4980  mapsn  6668