Theorem euabex 4085

Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
euabex	|- ( E! x ph -> { x | ph } e. _V )

Proof of Theorem euabex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3539	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		vex 2644	. . . . 5 |- y e. _V
3	2	snex 4049	. . . 4 |- { y } e. _V
4		eleq1 2162	. . . 4 |- ( { x | ph } = { y } -> ( { x | ph } e. _V <-> { y } e. _V ) )
5	3, 4	mpbiri 167	. . 3 |- ( { x | ph } = { y } -> { x | ph } e. _V )
6	5	exlimiv 1545	. 2 |- ( E. y { x | ph } = { y } -> { x | ph } e. _V )
7	1, 6	sylbi 120	1 |- ( E! x ph -> { x | ph } e. _V )

Syntax hints:   -> wi 4   = wceq 1299   E.wex 1436   e. wcel 1448   E!weu 1960   {cab 2086   _Vcvv 2641   {csn 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480

This theorem is referenced by: (None)