Theorem euabsn 3692

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 3691	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1542	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2341	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2349	. . 3 |- F/ x { x | ph } = { y }
5		sneq 3633	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2208	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 1770	. 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 1364   E.wex 1506   E!weu 2045   {cab 2182   {csn 3622

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by:  eusn  3696  args  5038  mapsn  6749