Theorem euabsn 3662

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 3661	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1528	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2321	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2329	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3603	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2189	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1756	. 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 1353   E.wex 1492   E!weu 2026   {cab 2163   {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sn 3598

This theorem is referenced by:  eusn  3666  args  4995  mapsn  6686