Theorem euind 3023

Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)

Hypotheses

Ref	Expression
euind.1	|- B e. _V
euind.2	|- (x = y -> (ph <-> ps))
euind.3	|- (x = y -> A = B)

Assertion

Ref	Expression
euind	|- ((A.xA.y((ph /\ ps) -> A = B) /\ E.xph) -> E!zA.x(ph -> z = A))

Distinct variable groups:   y,z,ph   x,z,ps   y,A,z   x,B,z   x,y

Allowed substitution hints:   ph(x)   ps(y)   A(x)   B(y)

Proof of Theorem euind

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euind.2	. . . . . 6 |- (x = y -> (ph <-> ps))
2	1	cbvexv 2003	. . . . 5 |- (E.xph <-> E.yps)
3		euind.1	. . . . . . . . 9 |- B e. _V
4	3	isseti 2865	. . . . . . . 8 |- E.z z = B
5	4	biantrur 492	. . . . . . 7 |- (ps <-> (E.z z = B /\ ps))
6	5	exbii 1582	. . . . . 6 |- (E.yps <-> E.y(E.z z = B /\ ps))
7		19.41v 1901	. . . . . . . 8 |- (E.z(z = B /\ ps) <-> (E.z z = B /\ ps))
8	7	exbii 1582	. . . . . . 7 |- (E.yE.z(z = B /\ ps) <-> E.y(E.z z = B /\ ps))
9		excom 1741	. . . . . . 7 |- (E.yE.z(z = B /\ ps) <-> E.zE.y(z = B /\ ps))
10	8, 9	bitr3i 242	. . . . . 6 |- (E.y(E.z z = B /\ ps) <-> E.zE.y(z = B /\ ps))
11	6, 10	bitri 240	. . . . 5 |- (E.yps <-> E.zE.y(z = B /\ ps))
12	2, 11	bitri 240	. . . 4 |- (E.xph <-> E.zE.y(z = B /\ ps))
13		eqeq2 2362	. . . . . . . . 9 |- (A = B -> (z = A <-> z = B))
14	13	imim2i 13	. . . . . . . 8 |- (((ph /\ ps) -> A = B) -> ((ph /\ ps) -> (z = A <-> z = B)))
15		bi2 189	. . . . . . . . . 10 |- ((z = A <-> z = B) -> (z = B -> z = A))
16	15	imim2i 13	. . . . . . . . 9 |- (((ph /\ ps) -> (z = A <-> z = B)) -> ((ph /\ ps) -> (z = B -> z = A)))
17		an31 775	. . . . . . . . . . 11 |- (((ph /\ ps) /\ z = B) <-> ((z = B /\ ps) /\ ph))
18	17	imbi1i 315	. . . . . . . . . 10 |- ((((ph /\ ps) /\ z = B) -> z = A) <-> (((z = B /\ ps) /\ ph) -> z = A))
19		impexp 433	. . . . . . . . . 10 |- ((((ph /\ ps) /\ z = B) -> z = A) <-> ((ph /\ ps) -> (z = B -> z = A)))
20		impexp 433	. . . . . . . . . 10 |- ((((z = B /\ ps) /\ ph) -> z = A) <-> ((z = B /\ ps) -> (ph -> z = A)))
21	18, 19, 20	3bitr3i 266	. . . . . . . . 9 |- (((ph /\ ps) -> (z = B -> z = A)) <-> ((z = B /\ ps) -> (ph -> z = A)))
22	16, 21	sylib 188	. . . . . . . 8 |- (((ph /\ ps) -> (z = A <-> z = B)) -> ((z = B /\ ps) -> (ph -> z = A)))
23	14, 22	syl 15	. . . . . . 7 |- (((ph /\ ps) -> A = B) -> ((z = B /\ ps) -> (ph -> z = A)))
24	23	2alimi 1560	. . . . . 6 |- (A.xA.y((ph /\ ps) -> A = B) -> A.xA.y((z = B /\ ps) -> (ph -> z = A)))
25		19.23v 1891	. . . . . . . 8 |- (A.y((z = B /\ ps) -> (ph -> z = A)) <-> (E.y(z = B /\ ps) -> (ph -> z = A)))
26	25	albii 1566	. . . . . . 7 |- (A.xA.y((z = B /\ ps) -> (ph -> z = A)) <-> A.x(E.y(z = B /\ ps) -> (ph -> z = A)))
27		19.21v 1890	. . . . . . 7 |- (A.x(E.y(z = B /\ ps) -> (ph -> z = A)) <-> (E.y(z = B /\ ps) -> A.x(ph -> z = A)))
28	26, 27	bitri 240	. . . . . 6 |- (A.xA.y((z = B /\ ps) -> (ph -> z = A)) <-> (E.y(z = B /\ ps) -> A.x(ph -> z = A)))
29	24, 28	sylib 188	. . . . 5 |- (A.xA.y((ph /\ ps) -> A = B) -> (E.y(z = B /\ ps) -> A.x(ph -> z = A)))
30	29	eximdv 1622	. . . 4 |- (A.xA.y((ph /\ ps) -> A = B) -> (E.zE.y(z = B /\ ps) -> E.zA.x(ph -> z = A)))
31	12, 30	syl5bi 208	. . 3 |- (A.xA.y((ph /\ ps) -> A = B) -> (E.xph -> E.zA.x(ph -> z = A)))
32	31	imp 418	. 2 |- ((A.xA.y((ph /\ ps) -> A = B) /\ E.xph) -> E.zA.x(ph -> z = A))
33		pm4.24 624	. . . . . . . 8 |- (ph <-> (ph /\ ph))
34	33	biimpi 186	. . . . . . 7 |- (ph -> (ph /\ ph))
35		prth 554	. . . . . . 7 |- (((ph -> z = A) /\ (ph -> w = A)) -> ((ph /\ ph) -> (z = A /\ w = A)))
36		eqtr3 2372	. . . . . . 7 |- ((z = A /\ w = A) -> z = w)
37	34, 35, 36	syl56 30	. . . . . 6 |- (((ph -> z = A) /\ (ph -> w = A)) -> (ph -> z = w))
38	37	alanimi 1562	. . . . 5 |- ((A.x(ph -> z = A) /\ A.x(ph -> w = A)) -> A.x(ph -> z = w))
39		19.23v 1891	. . . . . . 7 |- (A.x(ph -> z = w) <-> (E.xph -> z = w))
40	39	biimpi 186	. . . . . 6 |- (A.x(ph -> z = w) -> (E.xph -> z = w))
41	40	com12 27	. . . . 5 |- (E.xph -> (A.x(ph -> z = w) -> z = w))
42	38, 41	syl5 28	. . . 4 |- (E.xph -> ((A.x(ph -> z = A) /\ A.x(ph -> w = A)) -> z = w))
43	42	alrimivv 1632	. . 3 |- (E.xph -> A.zA.w((A.x(ph -> z = A) /\ A.x(ph -> w = A)) -> z = w))
44	43	adantl 452	. 2 |- ((A.xA.y((ph /\ ps) -> A = B) /\ E.xph) -> A.zA.w((A.x(ph -> z = A) /\ A.x(ph -> w = A)) -> z = w))
45		eqeq1 2359	. . . . 5 |- (z = w -> (z = A <-> w = A))
46	45	imbi2d 307	. . . 4 |- (z = w -> ((ph -> z = A) <-> (ph -> w = A)))
47	46	albidv 1625	. . 3 |- (z = w -> (A.x(ph -> z = A) <-> A.x(ph -> w = A)))
48	47	eu4 2243	. 2 |- (E!zA.x(ph -> z = A) <-> (E.zA.x(ph -> z = A) /\ A.zA.w((A.x(ph -> z = A) /\ A.x(ph -> w = A)) -> z = w)))
49	32, 44, 48	sylanbrc 645	1 |- ((A.xA.y((ph /\ ps) -> A = B) /\ E.xph) -> E!zA.x(ph -> z = A))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2859

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)