Theorem euabex 4311

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 3735	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		vex 2802	. . . . 5 |- y e. _V
3	2	snex 4269	. . . 4 |- { y } e. _V
4		eleq1 2292	. . . 4 |- ( { x | ph } = { y } -> ( { x | ph } e. _V <-> { y } e. _V ) )
5	3, 4	mpbiri 168	. . 3 |- ( { x | ph } = { y } -> { x | ph } e. _V )
6	5	exlimiv 1644	. 2 |- ( E. y { x | ph } = { y } -> { x | ph } e. _V )
7	1, 6	sylbi 121	1 |- ( E! x ph -> { x | ph } e. _V )

Syntax hints:   -> wi 4   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   {cab 2215   _Vcvv 2799   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by: (None)