Theorem euabsn 4261

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 4260	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv 1843	. . 3 |- F/ y { x | ph } = { x }
3		nfab1 2766	. . . 4 |- F/_ x { x | ph }
4	3	nfeq1 2778	. . 3 |- F/ x { x | ph } = { y }
5		sneq 4187	. . . 4 |- ( x = y -> { x } = { y } )
6	5	eqeq2d 2632	. . 3 |- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2, 4, 6	cbvex 2272	. 2 |- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1, 7	bitr4i 267	1 |- ( E! x ph <-> E. x { x | ph } = { x } )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   E!weu 2470   {cab 2608   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  eusn  4265  uniintsn  4514  args  5493  opabiotadm  6260  mapsn  7899  mapsnd  39388