Theorem moabex 3513

Description: "At most one" existence implies a class abstraction exists.

Assertion

Ref	Expression
moabex	|- (E*xph -> {x | ph} e. _V)

Proof of Theorem moabex

Step	Hyp	Ref	Expression
1		ax-17 1317	. . 3 |- (ph -> A.yph)
2	1	mo2 1795	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
3		hba1 1350	. . . . 5 |- (A.x(ph -> x = y) -> A.xA.x(ph -> x = y))
4		pm4.71 697	. . . . . . . 8 |- ((ph -> x = y) <-> (ph <-> (ph /\ x = y)))
5	4	biimpi 168	. . . . . . 7 |- ((ph -> x = y) -> (ph <-> (ph /\ x = y)))
6	5	a4s 1330	. . . . . 6 |- (A.x(ph -> x = y) -> (ph <-> (ph /\ x = y)))
7		ancom 482	. . . . . 6 |- ((ph /\ x = y) <-> (x = y /\ ph))
8	6, 7	syl6bb 595	. . . . 5 |- (A.x(ph -> x = y) -> (ph <-> (x = y /\ ph)))
9	3, 8	abbid 2007	. . . 4 |- (A.x(ph -> x = y) -> {x | ph} = {x | (x = y /\ ph)})
10		df-sn 3049	. . . . . 6 |- {y} = {x | x = y}
11		snex 3492	. . . . . 6 |- {y} e. _V
12	10, 11	eqeltrri 1968	. . . . 5 |- {x | x = y} e. _V
13		simpl 346	. . . . . 6 |- ((x = y /\ ph) -> x = y)
14	13	ss2abi 2679	. . . . 5 |- {x | (x = y /\ ph)} C_ {x | x = y}
15	12, 14	ssexi 3456	. . . 4 |- {x | (x = y /\ ph)} e. _V
16	9, 15	syl6eqel 1979	. . 3 |- (A.x(ph -> x = y) -> {x | ph} e. _V)
17	16	19.23aiv 1674	. 2 |- (E.yA.x(ph -> x = y) -> {x | ph} e. _V)
18	2, 17	sylbi 216	1 |- (E*xph -> {x | ph} e. _V)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E*wmo 1772  {cab 1871  _Vcvv 2292  {csn 3044

This theorem is referenced by:  euabex 3514  fvex 4689  supex 5667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049

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