Theorem euabsn 3713

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 3712	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1552	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2352	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2360	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3654	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2219	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1780	. 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 1373   E.wex 1516   E!weu 2055   {cab 2193   {csn 3643

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sn 3649

This theorem is referenced by:  eusn  3717  args  5070  mapsn  6800