Theorem mo 1020

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

Hypothesis

Ref	Expression
mo.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
mo	⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))

Distinct variable group(s):   x,y

Proof of Theorem mo

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

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃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

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