Theorem dfeu2 4334

Description: Alternate definition of existential uniqueness in terms of abstraction. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
dfeu2	|- (E!xph <-> {x | ph} e. 1c)

Proof of Theorem dfeu2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sn 3742	. . . . 5 |- {y} = {x | x = y}
2	1	eqeq2i 2363	. . . 4 |- ({x | ph} = {y} <-> {x | ph} = {x | x = y})
3		abbib 2464	. . . 4 |- ({x | ph} = {x | x = y} <-> A.x(ph <-> x = y))
4	2, 3	bitr2i 241	. . 3 |- (A.x(ph <-> x = y) <-> {x | ph} = {y})
5	4	exbii 1582	. 2 |- (E.yA.x(ph <-> x = y) <-> E.y{x | ph} = {y})
6		df-eu 2208	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
7		el1c 4140	. 2 |- ({x | ph} e. 1c <-> E.y{x | ph} = {y})
8	5, 6, 7	3bitr4i 268	1 |- (E!xph <-> {x | ph} e. 1c)

Syntax hints:   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339  {csn 3738  1_cc1c 4135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137

This theorem is referenced by:  euabex  4335  dfiota4  4373