Theorem euabex 4258

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 3691	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		vex 2766	. . . . 5 |- y e. _V
3	2	snex 4218	. . . 4 |- { y } e. _V
4		eleq1 2259	. . . 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 1612	. 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 1364   E.wex 1506   E!weu 2045   e. wcel 2167   {cab 2182   _Vcvv 2763   {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-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

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-in 3163  df-ss 3170  df-pw 3607  df-sn 3628

This theorem is referenced by: (None)