Theorem moabex 1868

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-17 925	. . 3 |- (ph -> A.yph)
2	1	mo2 1026	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
3		df-sn 1811	. . . . . 6 |- {y} = {x | x = y}
4		snex 1859	. . . . . 6 |- {y} e. V
5	3, 4	eqeltrr 1160	. . . . 5 |- {x | x = y} e. V
6		pm3.26 256	. . . . . 6 |- ((x = y /\ ph) -> x = y)
7	6	ss2abi 1552	. . . . 5 |- {x | (x = y /\ ph)} (_ {x | x = y}
8	5, 7	ssexi 1701	. . . 4 |- {x | (x = y /\ ph)} e. V
9		hba1 698	. . . . . 6 |- (A.x(ph -> x = y) -> A.xA.x(ph -> x = y))
10		pm4.71 481	. . . . . . . . 9 |- ((ph -> x = y) <-> (ph <-> (ph /\ x = y)))
11	10	biimp 133	. . . . . . . 8 |- ((ph -> x = y) -> (ph <-> (ph /\ x = y)))
12	11	a4s 682	. . . . . . 7 |- (A.x(ph -> x = y) -> (ph <-> (ph /\ x = y)))
13		ancom 333	. . . . . . 7 |- ((ph /\ x = y) <-> (x = y /\ ph))
14	12, 13	syl6bb 414	. . . . . 6 |- (A.x(ph -> x = y) -> (ph <-> (x = y /\ ph)))
15	9, 14	biabd 1182	. . . . 5 |- (A.x(ph -> x = y) -> {x | ph} = {x | (x = y /\ ph)})
16	15	eleq1d 1155	. . . 4 |- (A.x(ph -> x = y) -> ({x | ph} e. V <-> {x | (x = y /\ ph)} e. V))
17	8, 16	mpbiri 169	. . 3 |- (A.x(ph -> x = y) -> {x | ph} e. V)
18	17	19.23aiv 952	. 2 |- (E.yA.x(ph -> x = y) -> {x | ph} e. V)
19	2, 18	sylbi 174	1 |- (E*xph -> {x | ph} e. V)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E*wmo 1008  {cab 1090   e. wcel 1092  Vcvv 1348  {csn 1808

This theorem is referenced by:  euabex 1869  supex 2157  fvex 2838

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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