Theorem mo 1020

Description: Equivalent definitions of "there exists at most one".

Hypothesis

Ref	Expression
mo.1	|- (ph -> A.yph)

Assertion

Ref	Expression
mo	|- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Distinct variable group(s):   x,y

Proof of Theorem mo

Step	Hyp	Ref	Expression
1		mo.1	. . . . . 6 |- (ph -> A.yph)
2		ax-17 925	. . . . . 6 |- (x = z -> A.y x = z)
3	1, 2	hbim 702	. . . . 5 |- ((ph -> x = z) -> A.y(ph -> x = z))
4	3	hbal 700	. . . 4 |- (A.x(ph -> x = z) -> A.yA.x(ph -> x = z))
5		ax-17 925	. . . 4 |- (A.x(ph -> x = y) -> A.zA.x(ph -> x = y))
6		eqt2b 818	. . . . . 6 |- (z = y -> (x = z <-> x = y))
7	6	imbi2d 464	. . . . 5 |- (z = y -> ((ph -> x = z) <-> (ph -> x = y)))
8	7	bialdv 935	. . . 4 |- (z = y -> (A.x(ph -> x = z) <-> A.x(ph -> x = y)))
9	4, 5, 8	cbvex 849	. . 3 |- (E.zA.x(ph -> x = z) <-> E.yA.x(ph -> x = y))
10		hbs1 986	. . . . . . . . 9 |- ([y / x]ph -> A.x[y / x]ph)
11		ax-17 925	. . . . . . . . 9 |- (y = z -> A.x y = z)
12	10, 11	hbim 702	. . . . . . . 8 |- (([y / x]ph -> y = z) -> A.x([y / x]ph -> y = z))
13		sbequ2 864	. . . . . . . . 9 |- (x = y -> ([y / x]ph -> ph))
14		ax-8 798	. . . . . . . . 9 |- (x = y -> (x = z -> y = z))
15	13, 14	syl34d 29	. . . . . . . 8 |- (x = y -> ((ph -> x = z) -> ([y / x]ph -> y = z)))
16	3, 12, 15	cbv3 847	. . . . . . 7 |- (A.x(ph -> x = z) -> A.y([y / x]ph -> y = z))
17	16	ancli 244	. . . . . 6 |- (A.x(ph -> x = z) -> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
18	3, 12	aaan 794	. . . . . 6 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) <-> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
19	17, 18	sylibr 175	. . . . 5 |- (A.x(ph -> x = z) -> A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)))
20		prth 429	. . . . . . . 8 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> (x = z /\ y = z)))
21		eqan 816	. . . . . . . 8 |- ((x = z /\ y = z) -> x = y)
22	20, 21	syl6 23	. . . . . . 7 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> x = y))
23	22	19.20i 691	. . . . . 6 |- (A.y((ph -> x = z) /\ ([y / x]ph -> y = z)) -> A.y((ph /\ [y / x]ph) -> x = y))
24	23	19.20i 691	. . . . 5 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
25	19, 24	syl 12	. . . 4 |- (A.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
26	25	19.23aiv 952	. . 3 |- (E.zA.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
27	9, 26	sylbir 176	. 2 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
28	1	hbsb3 875	. . . . . 6 |- ([y / x]ph -> A.x[y / x]ph)
29	28	19.22i 723	. . . . 5 |- (E.y[y / x]ph -> E.yA.x[y / x]ph)
30		19.20 690	. . . . . . . . 9 |- (A.x([y / x]ph -> (ph -> x = y)) -> (A.x[y / x]ph -> A.x(ph -> x = y)))
31	30	19.20i 691	. . . . . . . 8 |- (A.yA.x([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
32	31	a7s 689	. . . . . . 7 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
33		19.22 722	. . . . . . 7 |- (A.y(A.x[y / x]ph -> A.x(ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
34	32, 33	syl 12	. . . . . 6 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
35	34	com12 13	. . . . 5 |- (E.yA.x[y / x]ph -> (A.xA.y([y / x]ph -> (ph -> x = y)) -> E.yA.x(ph -> x = y)))
36	29, 35	syl 12	. . . 4 |- (E.y[y / x]ph -> (A.xA.y([y / x]ph -> (ph -> x = y)) -> E.yA.x(ph -> x = y)))
37		impexp 276	. . . . . 6 |- (((ph /\ [y / x]ph) -> x = y) <-> (ph -> ([y / x]ph -> x = y)))
38		bi2.04 141	. . . . . 6 |- ((ph -> ([y / x]ph -> x = y)) <-> ([y / x]ph -> (ph -> x = y)))
39	37, 38	bitr 151	. . . . 5 |- (((ph /\ [y / x]ph) -> x = y) <-> ([y / x]ph -> (ph -> x = y)))
40	39	bi2al 696	. . . 4 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) <-> A.xA.y([y / x]ph -> (ph -> x = y)))
41	36, 40	syl5ib 181	. . 3 |- (E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
42		alnex 716	. . . . 5 |- (A.y -. [y / x]ph <-> -. E.y[y / x]ph)
43	28	hbne 699	. . . . . . 7 |- (-. [y / x]ph -> A.x -. [y / x]ph)
44	1	hbne 699	. . . . . . 7 |- (-. ph -> A.y -. ph)
45		sbequ1 863	. . . . . . . . 9 |- (x = y -> (ph -> [y / x]ph))
46	45	eqcoms 813	. . . . . . . 8 |- (y = x -> (ph -> [y / x]ph))
47	46	con3d 87	. . . . . . 7 |- (y = x -> (-. [y / x]ph -> -. ph))
48	43, 44, 47	cbv3 847	. . . . . 6 |- (A.y -. [y / x]ph -> A.x -. ph)
49		pm2.21 71	. . . . . . 7 |- (-. ph -> (ph -> x = y))
50	49	19.20i 691	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
51		19.8a 712	. . . . . 6 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
52	48, 50, 51	3syl 21	. . . . 5 |- (A.y -. [y / x]ph -> E.yA.x(ph -> x = y))
53	42, 52	sylbir 176	. . . 4 |- (-. E.y[y / x]ph -> E.yA.x(ph -> x = y))
54	53	a1d 14	. . 3 |- (-. E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
55	41, 54	pm2.61i 110	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y))
56	27, 55	impbi 139	1 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852

This theorem is referenced by:  eu2 1023  eu3 1024  mo3 1027

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-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

metamath.org