Theorem euind 3717

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))

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	cbvexvw 2044	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
3		euind.1	. . . . . . . . 9 ⊢ 𝐵 ∈ V
4	3	isseti 3510	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝐵
5	4	biantrur 533	. . . . . . 7 ⊢ (𝜓 ↔ (∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
6	5	exbii 1848	. . . . . 6 ⊢ (∃𝑦𝜓 ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
7		19.41v 1950	. . . . . . 7 ⊢ (∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ (∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
8	7	exbii 1848	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵 ∧ 𝜓))
9		excom 2169	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑧 = 𝐵 ∧ 𝜓) ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
10	6, 8, 9	3bitr2i 301	. . . . 5 ⊢ (∃𝑦𝜓 ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
11	2, 10	bitri 277	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓))
12		eqeq2 2835	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵))
13	12	imim2i 16	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
14		biimpr 222	. . . . . . . . . 10 ⊢ ((𝑧 = 𝐴 ↔ 𝑧 = 𝐵) → (𝑧 = 𝐵 → 𝑧 = 𝐴))
15	14	imim2i 16	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)) → ((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)))
16		an31 646	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) ↔ ((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑))
17	16	imbi1i 352	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ (((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑) → 𝑧 = 𝐴))
18		impexp 453	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ ((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)))
19		impexp 453	. . . . . . . . . 10 ⊢ ((((𝑧 = 𝐵 ∧ 𝜓) ∧ 𝜑) → 𝑧 = 𝐴) ↔ ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
20	17, 18, 19	3bitr3i 303	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐵 → 𝑧 = 𝐴)) ↔ ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
21	15, 20	sylib 220	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)) → ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
22	13, 21	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
23	22	2alimi 1813	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → ∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
24		19.23v 1943	. . . . . . . 8 ⊢ (∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
25	24	albii 1820	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ ∀𝑥(∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)))
26		19.21v 1940	. . . . . . 7 ⊢ (∀𝑥(∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
27	25, 26	bitri 277	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑧 = 𝐵 ∧ 𝜓) → (𝜑 → 𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
28	23, 27	sylib 220	. . . . 5 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∀𝑥(𝜑 → 𝑧 = 𝐴)))
29	28	eximdv 1918	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑧∃𝑦(𝑧 = 𝐵 ∧ 𝜓) → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)))
30	11, 29	syl5bi 244	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) → (∃𝑥𝜑 → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)))
31	30	imp 409	. 2 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))
32		pm4.24 566	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
33	32	biimpi 218	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ 𝜑))
34		anim12 807	. . . . . . . 8 ⊢ (((𝜑 → 𝑧 = 𝐴) ∧ (𝜑 → 𝑤 = 𝐴)) → ((𝜑 ∧ 𝜑) → (𝑧 = 𝐴 ∧ 𝑤 = 𝐴)))
35		eqtr3 2845	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐴) → 𝑧 = 𝑤)
36	33, 34, 35	syl56 36	. . . . . . 7 ⊢ (((𝜑 → 𝑧 = 𝐴) ∧ (𝜑 → 𝑤 = 𝐴)) → (𝜑 → 𝑧 = 𝑤))
37	36	alanimi 1817	. . . . . 6 ⊢ ((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → ∀𝑥(𝜑 → 𝑧 = 𝑤))
38		19.23v 1943	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑧 = 𝑤) ↔ (∃𝑥𝜑 → 𝑧 = 𝑤))
39	37, 38	sylib 220	. . . . 5 ⊢ ((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → (∃𝑥𝜑 → 𝑧 = 𝑤))
40	39	com12 32	. . . 4 ⊢ (∃𝑥𝜑 → ((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
41	40	alrimivv 1929	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
42	41	adantl 484	. 2 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤))
43		eqeq1 2827	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝐴 ↔ 𝑤 = 𝐴))
44	43	imbi2d 343	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝜑 → 𝑧 = 𝐴) ↔ (𝜑 → 𝑤 = 𝐴)))
45	44	albidv 1921	. . 3 ⊢ (𝑧 = 𝑤 → (∀𝑥(𝜑 → 𝑧 = 𝐴) ↔ ∀𝑥(𝜑 → 𝑤 = 𝐴)))
46	45	eu4 2699	. 2 ⊢ (∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴) ↔ (∃𝑧∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑧∀𝑤((∀𝑥(𝜑 → 𝑧 = 𝐴) ∧ ∀𝑥(𝜑 → 𝑤 = 𝐴)) → 𝑧 = 𝑤)))
47	31, 42, 46	sylanbrc 585	1 ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  Vcvv 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-mo 2622  df-eu 2654  df-cleq 2816  df-clel 2895

This theorem is referenced by: (None)