Theorem euind 3225

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. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E! z A. 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 2117	. . . . 5 |- ( E. x ph <-> E. y ps )
3		euind.1	. . . . . . . . 9 |- B e. _V
4	3	isseti 3051	. . . . . . . 8 |- E. z z = B
5	4	biantrur 509	. . . . . . 7 |- ( ps <-> ( E. z z = B /\ ps ) )
6	5	exbii 1718	. . . . . 6 |- ( E. y ps <-> E. y ( E. z z = B /\ ps ) )
7		19.41v 1830	. . . . . . 7 |- ( E. z ( z = B /\ ps ) <-> ( E. z z = B /\ ps ) )
8	7	exbii 1718	. . . . . 6 |- ( E. y E. z ( z = B /\ ps ) <-> E. y ( E. z z = B /\ ps ) )
9		excom 1927	. . . . . 6 |- ( E. y E. z ( z = B /\ ps ) <-> E. z E. y ( z = B /\ ps ) )
10	6, 8, 9	3bitr2i 277	. . . . 5 |- ( E. y ps <-> E. z E. y ( z = B /\ ps ) )
11	2, 10	bitri 253	. . . 4 |- ( E. x ph <-> E. z E. y ( z = B /\ ps ) )
12		eqeq2 2462	. . . . . . . . 9 |- ( A = B -> ( z = A <-> z = B ) )
13	12	imim2i 16	. . . . . . . 8 |- ( ( ( ph /\ ps ) -> A = B ) -> ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) )
14		biimpr 202	. . . . . . . . . 10 |- ( ( z = A <-> z = B ) -> ( z = B -> z = A ) )
15	14	imim2i 16	. . . . . . . . 9 |- ( ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) -> ( ( ph /\ ps ) -> ( z = B -> z = A ) ) )
16		an31 809	. . . . . . . . . . 11 |- ( ( ( ph /\ ps ) /\ z = B ) <-> ( ( z = B /\ ps ) /\ ph ) )
17	16	imbi1i 327	. . . . . . . . . 10 |- ( ( ( ( ph /\ ps ) /\ z = B ) -> z = A ) <-> ( ( ( z = B /\ ps ) /\ ph ) -> z = A ) )
18		impexp 448	. . . . . . . . . 10 |- ( ( ( ( ph /\ ps ) /\ z = B ) -> z = A ) <-> ( ( ph /\ ps ) -> ( z = B -> z = A ) ) )
19		impexp 448	. . . . . . . . . 10 |- ( ( ( ( z = B /\ ps ) /\ ph ) -> z = A ) <-> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
20	17, 18, 19	3bitr3i 279	. . . . . . . . 9 |- ( ( ( ph /\ ps ) -> ( z = B -> z = A ) ) <-> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
21	15, 20	sylib 200	. . . . . . . 8 |- ( ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) -> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
22	13, 21	syl 17	. . . . . . 7 |- ( ( ( ph /\ ps ) -> A = B ) -> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
23	22	2alimi 1685	. . . . . 6 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> A. x A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
24		19.23v 1818	. . . . . . . 8 |- ( A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> ( E. y ( z = B /\ ps ) -> ( ph -> z = A ) ) )
25	24	albii 1691	. . . . . . 7 |- ( A. x A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> A. x ( E. y ( z = B /\ ps ) -> ( ph -> z = A ) ) )
26		19.21v 1786	. . . . . . 7 |- ( A. x ( E. y ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> ( E. y ( z = B /\ ps ) -> A. x ( ph -> z = A ) ) )
27	25, 26	bitri 253	. . . . . 6 |- ( A. x A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> ( E. y ( z = B /\ ps ) -> A. x ( ph -> z = A ) ) )
28	23, 27	sylib 200	. . . . 5 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> ( E. y ( z = B /\ ps ) -> A. x ( ph -> z = A ) ) )
29	28	eximdv 1764	. . . 4 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> ( E. z E. y ( z = B /\ ps ) -> E. z A. x ( ph -> z = A ) ) )
30	11, 29	syl5bi 221	. . 3 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> ( E. x ph -> E. z A. x ( ph -> z = A ) ) )
31	30	imp 431	. 2 |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E. z A. x ( ph -> z = A ) )
32		pm4.24 649	. . . . . . . 8 |- ( ph <-> ( ph /\ ph ) )
33	32	biimpi 198	. . . . . . 7 |- ( ph -> ( ph /\ ph ) )
34		prth 575	. . . . . . 7 |- ( ( ( ph -> z = A ) /\ ( ph -> w = A ) ) -> ( ( ph /\ ph ) -> ( z = A /\ w = A ) ) )
35		eqtr3 2472	. . . . . . 7 |- ( ( z = A /\ w = A ) -> z = w )
36	33, 34, 35	syl56 35	. . . . . 6 |- ( ( ( ph -> z = A ) /\ ( ph -> w = A ) ) -> ( ph -> z = w ) )
37	36	alanimi 1688	. . . . 5 |- ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> A. x ( ph -> z = w ) )
38		19.23v 1818	. . . . . . 7 |- ( A. x ( ph -> z = w ) <-> ( E. x ph -> z = w ) )
39	38	biimpi 198	. . . . . 6 |- ( A. x ( ph -> z = w ) -> ( E. x ph -> z = w ) )
40	39	com12 32	. . . . 5 |- ( E. x ph -> ( A. x ( ph -> z = w ) -> z = w ) )
41	37, 40	syl5 33	. . . 4 |- ( E. x ph -> ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) )
42	41	alrimivv 1774	. . 3 |- ( E. x ph -> A. z A. w ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) )
43	42	adantl 468	. 2 |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> A. z A. w ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) )
44		eqeq1 2455	. . . . 5 |- ( z = w -> ( z = A <-> w = A ) )
45	44	imbi2d 318	. . . 4 |- ( z = w -> ( ( ph -> z = A ) <-> ( ph -> w = A ) ) )
46	45	albidv 1767	. . 3 |- ( z = w -> ( A. x ( ph -> z = A ) <-> A. x ( ph -> w = A ) ) )
47	46	eu4 2347	. 2 |- ( E! z A. x ( ph -> z = A ) <-> ( E. z A. x ( ph -> z = A ) /\ A. z A. w ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) ) )
48	31, 43, 47	sylanbrc 670	1 |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E! z A. x ( ph -> z = A ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   E!weu 2299   _Vcvv 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3047

This theorem is referenced by: (None)