Theorem euabex 4203

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 3645	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		vex 2729	. . . . 5 |- y e. _V
3	2	snex 4164	. . . 4 |- { y } e. _V
4		eleq1 2229	. . . 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 1586	. 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 1343   E.wex 1480   E!weu 2014   e. wcel 2136   {cab 2151   _Vcvv 2726   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582

This theorem is referenced by: (None)