Theorem euind 2924

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 1918	. . . . 5 |- ( E. x ph <-> E. y ps )
3		euind.1	. . . . . . . . 9 |- B e. _V
4	3	isseti 2745	. . . . . . . 8 |- E. z z = B
5	4	biantrur 303	. . . . . . 7 |- ( ps <-> ( E. z z = B /\ ps ) )
6	5	exbii 1605	. . . . . 6 |- ( E. y ps <-> E. y ( E. z z = B /\ ps ) )
7		19.41v 1902	. . . . . . 7 |- ( E. z ( z = B /\ ps ) <-> ( E. z z = B /\ ps ) )
8	7	exbii 1605	. . . . . 6 |- ( E. y E. z ( z = B /\ ps ) <-> E. y ( E. z z = B /\ ps ) )
9		excom 1664	. . . . . 6 |- ( E. y E. z ( z = B /\ ps ) <-> E. z E. y ( z = B /\ ps ) )
10	6, 8, 9	3bitr2i 208	. . . . 5 |- ( E. y ps <-> E. z E. y ( z = B /\ ps ) )
11	2, 10	bitri 184	. . . 4 |- ( E. x ph <-> E. z E. y ( z = B /\ ps ) )
12		eqeq2 2187	. . . . . . . . 9 |- ( A = B -> ( z = A <-> z = B ) )
13	12	imim2i 12	. . . . . . . 8 |- ( ( ( ph /\ ps ) -> A = B ) -> ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) )
14		biimpr 130	. . . . . . . . . 10 |- ( ( z = A <-> z = B ) -> ( z = B -> z = A ) )
15	14	imim2i 12	. . . . . . . . 9 |- ( ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) -> ( ( ph /\ ps ) -> ( z = B -> z = A ) ) )
16		an31 564	. . . . . . . . . . 11 |- ( ( ( ph /\ ps ) /\ z = B ) <-> ( ( z = B /\ ps ) /\ ph ) )
17	16	imbi1i 238	. . . . . . . . . 10 |- ( ( ( ( ph /\ ps ) /\ z = B ) -> z = A ) <-> ( ( ( z = B /\ ps ) /\ ph ) -> z = A ) )
18		impexp 263	. . . . . . . . . 10 |- ( ( ( ( ph /\ ps ) /\ z = B ) -> z = A ) <-> ( ( ph /\ ps ) -> ( z = B -> z = A ) ) )
19		impexp 263	. . . . . . . . . 10 |- ( ( ( ( z = B /\ ps ) /\ ph ) -> z = A ) <-> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
20	17, 18, 19	3bitr3i 210	. . . . . . . . 9 |- ( ( ( ph /\ ps ) -> ( z = B -> z = A ) ) <-> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
21	15, 20	sylib 122	. . . . . . . 8 |- ( ( ( ph /\ ps ) -> ( z = A <-> z = B ) ) -> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
22	13, 21	syl 14	. . . . . . 7 |- ( ( ( ph /\ ps ) -> A = B ) -> ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
23	22	2alimi 1456	. . . . . 6 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> A. x A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) )
24		19.23v 1883	. . . . . . . 8 |- ( A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> ( E. y ( z = B /\ ps ) -> ( ph -> z = A ) ) )
25	24	albii 1470	. . . . . . 7 |- ( A. x A. y ( ( z = B /\ ps ) -> ( ph -> z = A ) ) <-> A. x ( E. y ( z = B /\ ps ) -> ( ph -> z = A ) ) )
26		19.21v 1873	. . . . . . 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 184	. . . . . 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 122	. . . . 5 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> ( E. y ( z = B /\ ps ) -> A. x ( ph -> z = A ) ) )
29	28	eximdv 1880	. . . 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	biimtrid 152	. . 3 |- ( A. x A. y ( ( ph /\ ps ) -> A = B ) -> ( E. x ph -> E. z A. x ( ph -> z = A ) ) )
31	30	imp 124	. 2 |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E. z A. x ( ph -> z = A ) )
32		pm4.24 395	. . . . . . . 8 |- ( ph <-> ( ph /\ ph ) )
33	32	biimpi 120	. . . . . . 7 |- ( ph -> ( ph /\ ph ) )
34		anim12 344	. . . . . . 7 |- ( ( ( ph -> z = A ) /\ ( ph -> w = A ) ) -> ( ( ph /\ ph ) -> ( z = A /\ w = A ) ) )
35		eqtr3 2197	. . . . . . 7 |- ( ( z = A /\ w = A ) -> z = w )
36	33, 34, 35	syl56 34	. . . . . 6 |- ( ( ( ph -> z = A ) /\ ( ph -> w = A ) ) -> ( ph -> z = w ) )
37	36	alanimi 1459	. . . . 5 |- ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> A. x ( ph -> z = w ) )
38		19.23v 1883	. . . . . . 7 |- ( A. x ( ph -> z = w ) <-> ( E. x ph -> z = w ) )
39	38	biimpi 120	. . . . . 6 |- ( A. x ( ph -> z = w ) -> ( E. x ph -> z = w ) )
40	39	com12 30	. . . . 5 |- ( E. x ph -> ( A. x ( ph -> z = w ) -> z = w ) )
41	37, 40	syl5 32	. . . 4 |- ( E. x ph -> ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) )
42	41	alrimivv 1875	. . 3 |- ( E. x ph -> A. z A. w ( ( A. x ( ph -> z = A ) /\ A. x ( ph -> w = A ) ) -> z = w ) )
43	42	adantl 277	. 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 2184	. . . . 5 |- ( z = w -> ( z = A <-> w = A ) )
45	44	imbi2d 230	. . . 4 |- ( z = w -> ( ( ph -> z = A ) <-> ( ph -> w = A ) ) )
46	45	albidv 1824	. . 3 |- ( z = w -> ( A. x ( ph -> z = A ) <-> A. x ( ph -> w = A ) ) )
47	46	eu4 2088	. 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 417	1 |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E! z A. x ( ph -> z = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   E!weu 2026   e. wcel 2148   _Vcvv 2737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739

This theorem is referenced by: (None)