Theorem euabex 4269

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 3702	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		vex 2775	. . . . 5 |- y e. _V
3	2	snex 4229	. . . 4 |- { y } e. _V
4		eleq1 2268	. . . 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 1621	. 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 1373   E.wex 1515   E!weu 2054   e. wcel 2176   {cab 2191   _Vcvv 2772   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639

This theorem is referenced by: (None)