Theorem euabsn 3674

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 3673	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1538	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2331	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2339	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3615	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2199	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1766	. 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 1363   E.wex 1502   E!weu 2036   {cab 2173   {csn 3604

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sn 3610

This theorem is referenced by:  eusn  3678  args  5009  mapsn  6703