Theorem euind 2951

Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)

Hypotheses

Ref	Expression
euind.1	⊢ 𝐵 ∈ V
euind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
euind.3	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
euind	⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))

Distinct variable groups:   𝑦,𝑧,𝜑   𝑥,𝑧,𝜓   𝑦,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(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 → (𝜑 ↔ 𝜓))
2	1	cbvexv 1933	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
3		euind.1	. . . . . . . . 9 ⊢ 𝐵 ∈ V
4	3	isseti 2771	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝐵
5	4	biantrur 303	. . . . . . 7 ⊢ (𝜓 ↔ (∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
6	5	exbii 1619	. . . . . 6 ⊢ (∃𝑦𝜓 ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
7		19.41v 1917	. . . . . . 7 ⊢ (∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ (∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
8	7	exbii 1619	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
9		excom 1678	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
10	6, 8, 9	3bitr2i 208	. . . . 5 ⊢ (∃𝑦𝜓 ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
11	2, 10	bitri 184	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
12		eqeq2 2206	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵))
13	12	imim2i 12	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
14		biimpr 130	. . . . . . . . . 10 ⊢ ((𝑧 = 𝐴 ↔ 𝑧 = 𝐵) → (𝑧 = 𝐵 → 𝑧 = 𝐴))
15	14	imim2i 12	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)) → ((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)))
16		an31 564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) ↔ ((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑))
17	16	imbi1i 238	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ (((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑) → 𝑧 = 𝐴))
18		impexp 263	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ ((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)))
19		impexp 263	. . . . . . . . . 10 ⊢ ((((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑) → 𝑧 = 𝐴) ↔ ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
20	17, 18, 19	3bitr3i 210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)) ↔ ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
21	15, 20	sylib 122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)) → ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
22	13, 21	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
23	22	2alimi 1470	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
24		19.23v 1897	. . . . . . . 8 ⊢ (∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
25	24	albii 1484	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ ∀𝑥(∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
26		19.21v 1887	. . . . . . 7 ⊢ (∀𝑥(∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
27	25, 26	bitri 184	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
28	23, 27	sylib 122	. . . . 5 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
29	28	eximdv 1894	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)))
30	11, 29	biimtrid 152	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑥𝜑 → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)))
31	30	imp 124	. 2 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))
32		pm4.24 395	. . . . . . . 8 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
33	32	biimpi 120	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ 𝜑))
34		anim12 344	. . . . . . 7 ⊢ (((𝜑 → 𝑧 = 𝐴) ∧ (𝜑 → 𝑤 = 𝐴)) → ((𝜑 ∧ 𝜑) → (𝑧 = 𝐴 ∧ 𝑤 = 𝐴)))
35		eqtr3 2216	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐴) → 𝑧 = 𝑤)
36	33, 34, 35	syl56 34	. . . . . 6 ⊢ (((𝜑 → 𝑧 = 𝐴) ∧ (𝜑 → 𝑤 = 𝐴)) → (𝜑 → 𝑧 = 𝑤))
37	36	alanimi 1473	. . . . 5 ⊢ ((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → ∀𝑥(𝜑 → 𝑧 = 𝑤))
38		19.23v 1897	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑧 = 𝑤) ↔ (∃𝑥𝜑 → 𝑧 = 𝑤))
39	38	biimpi 120	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑧 = 𝑤) → (∃𝑥𝜑 → 𝑧 = 𝑤))
40	39	com12 30	. . . . 5 ⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝑧 = 𝑤) → 𝑧 = 𝑤))
41	37, 40	syl5 32	. . . 4 ⊢ (∃𝑥𝜑 → ((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
42	41	alrimivv 1889	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
43	42	adantl 277	. 2 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
44		eqeq1 2203	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝐴 ↔ 𝑤 = 𝐴))
45	44	imbi2d 230	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝜑 → 𝑧 = 𝐴) ↔ (𝜑 → 𝑤 = 𝐴)))
46	45	albidv 1838	. . 3 ⊢ (𝑧 = 𝑤 → (∀𝑥(𝜑 → 𝑧 = 𝐴) ↔ ∀𝑥(𝜑 → 𝑤 = 𝐴)))
47	46	eu4 2107	. 2 ⊢ (∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴) ↔ (∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤)))
48	31, 43, 47	sylanbrc 417	1 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1506  ∃!weu 2045   ∈ wcel 2167  Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by: (None)