Theorem mo 1386

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:   x,y

Proof of Theorem mo

Step	Hyp	Ref	Expression
1		mo.1	. . . . . 6 |- (ph -> A.yph)
2		ax-17 968	. . . . . 6 |- (x = z -> A.y x = z)
3	1, 2	hbim 1004	. . . . 5 |- ((ph -> x = z) -> A.y(ph -> x = z))
4	3	hbal 1002	. . . 4 |- (A.x(ph -> x = z) -> A.yA.x(ph -> x = z))
5		ax-17 968	. . . 4 |- (A.x(ph -> x = y) -> A.zA.x(ph -> x = y))
6		equequ2 1131	. . . . . 6 |- (z = y -> (x = z <-> x = y))
7	6	imbi2d 610	. . . . 5 |- (z = y -> ((ph -> x = z) <-> (ph -> x = y)))
8	7	albidv 1273	. . . 4 |- (z = y -> (A.x(ph -> x = z) <-> A.x(ph -> x = y)))
9	4, 5, 8	cbvex 1162	. . 3 |- (E.zA.x(ph -> x = z) <-> E.yA.x(ph -> x = y))
10		hbs1 1327	. . . . . . . . 9 |- ([y / x]ph -> A.x[y / x]ph)
11		ax-17 968	. . . . . . . . 9 |- (y = z -> A.x y = z)
12	10, 11	hbim 1004	. . . . . . . 8 |- (([y / x]ph -> y = z) -> A.x([y / x]ph -> y = z))
13		sbequ2 1175	. . . . . . . . 9 |- (x = y -> ([y / x]ph -> ph))
14		ax-8 961	. . . . . . . . 9 |- (x = y -> (x = z -> y = z))
15	13, 14	imim12d 29	. . . . . . . 8 |- (x = y -> ((ph -> x = z) -> ([y / x]ph -> y = z)))
16	3, 12, 15	cbv3 1160	. . . . . . 7 |- (A.x(ph -> x = z) -> A.y([y / x]ph -> y = z))
17	16	ancli 296	. . . . . 6 |- (A.x(ph -> x = z) -> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
18	3, 12	aaan 1115	. . . . . 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 200	. . . . 5 |- (A.x(ph -> x = z) -> A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)))
20		prth 554	. . . . . . 7 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> (x = z /\ y = z)))
21		equtr2 1129	. . . . . . 7 |- ((x = z /\ y = z) -> x = y)
22	20, 21	syl6 22	. . . . . 6 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> x = y))
23	22	19.20i2 990	. . . . 5 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
24	19, 23	syl 10	. . . 4 |- (A.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
25	24	19.23aiv 1290	. . 3 |- (E.zA.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
26	9, 25	sylbir 201	. 2 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
27		19.20 991	. . . . . . . 8 |- (A.x([y / x]ph -> (ph -> x = y)) -> (A.x[y / x]ph -> A.x(ph -> x = y)))
28	27	19.20i 989	. . . . . . 7 |- (A.yA.x([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
29	28	a7s 988	. . . . . 6 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
30		19.22 1035	. . . . . 6 |- (A.y(A.x[y / x]ph -> A.x(ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
31	29, 30	syl 10	. . . . 5 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
32	1	hbsb3 1202	. . . . . 6 |- ([y / x]ph -> A.x[y / x]ph)
33	32	19.22i 1036	. . . . 5 |- (E.y[y / x]ph -> E.yA.x[y / x]ph)
34	31, 33	syl5com 52	. . . 4 |- (E.y[y / x]ph -> (A.xA.y([y / x]ph -> (ph -> x = y)) -> E.yA.x(ph -> x = y)))
35		impexp 347	. . . . . 6 |- (((ph /\ [y / x]ph) -> x = y) <-> (ph -> ([y / x]ph -> x = y)))
36		bi2.04 160	. . . . . 6 |- ((ph -> ([y / x]ph -> x = y)) <-> ([y / x]ph -> (ph -> x = y)))
37	35, 36	bitr 173	. . . . 5 |- (((ph /\ [y / x]ph) -> x = y) <-> ([y / x]ph -> (ph -> x = y)))
38	37	2albii 997	. . . 4 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) <-> A.xA.y([y / x]ph -> (ph -> x = y)))
39	34, 38	syl5ib 206	. . 3 |- (E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
40		alnex 1029	. . . . 5 |- (A.y -. [y / x]ph <-> -. E.y[y / x]ph)
41	32	hbn 1001	. . . . . . 7 |- (-. [y / x]ph -> A.x -. [y / x]ph)
42	1	hbn 1001	. . . . . . 7 |- (-. ph -> A.y -. ph)
43		sbequ1 1174	. . . . . . . . 9 |- (x = y -> (ph -> [y / x]ph))
44	43	equcoms 1126	. . . . . . . 8 |- (y = x -> (ph -> [y / x]ph))
45	44	con3d 95	. . . . . . 7 |- (y = x -> (-. [y / x]ph -> -. ph))
46	41, 42, 45	cbv3 1160	. . . . . 6 |- (A.y -. [y / x]ph -> A.x -. ph)
47		pm2.21 76	. . . . . . 7 |- (-. ph -> (ph -> x = y))
48	47	19.20i 989	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
49		19.8a 1025	. . . . . 6 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
50	46, 48, 49	3syl 20	. . . . 5 |- (A.y -. [y / x]ph -> E.yA.x(ph -> x = y))
51	40, 50	sylbir 201	. . . 4 |- (-. E.y[y / x]ph -> E.yA.x(ph -> x = y))
52	51	a1d 12	. . 3 |- (-. E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
53	39, 52	pm2.61i 126	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y))
54	26, 53	impbi 157	1 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  [wsbc 1166

This theorem is referenced by:  eu2 1389  eu3 1390  mo3 1394

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168

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