Theorem euabsn 3440

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 3439	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1421	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2180	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2187	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3386	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2051	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1639	. 2 |- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1, 7	bitr4i 176	1 |- ( E! x ph <-> E. x { x | ph } = { x } )

Syntax hints:   <-> wb 98   = wceq 1243   E.wex 1381   E!weu 1900   {cab 2026   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  eusn  3444  args  4694