Theorem reuprg0 4618

Description: Convert a restricted existential uniqueness over a pair to a disjunction of conjunctions. (Contributed by AV, 2-Apr-2023.)

Hypotheses

Ref	Expression
reuprg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
reuprg.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
reuprg0	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem reuprg0

Dummy variables 𝑤 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsbc1v 3714	. . 3 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜑
2		nfsbc1v 3714	. . 3 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
3		sbceq1a 3705	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
4		dfsbcq 3696	. . 3 ⊢ (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑 ↔ [𝑐 / 𝑥]𝜑))
5	1, 2, 3, 4	reu8nf 3789	. 2 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)))
6		nfv 1922	. . . . 5 ⊢ Ⅎ𝑥𝜓
7		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑐
9	1, 8	nfim 1904	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
10	7, 9	nfralw 3147	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
11	6, 10	nfan 1907	. . . 4 ⊢ Ⅎ𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))
12		nfv 1922	. . . . 5 ⊢ Ⅎ𝑥𝜒
13		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑥 𝐵 = 𝑐
14	1, 13	nfim 1904	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)
15	7, 14	nfralw 3147	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)
16	12, 15	nfan 1907	. . . 4 ⊢ Ⅎ𝑥(𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))
17		reuprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
18		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑐 ↔ 𝐴 = 𝑐))
19	18	imbi2d 344	. . . . . 6 ⊢ (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
20	19	ralbidv 3118	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
21	17, 20	anbi12d 634	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))))
22		reuprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
23		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑐 ↔ 𝐵 = 𝑐))
24	23	imbi2d 344	. . . . . 6 ⊢ (𝑥 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))
25	24	ralbidv 3118	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))
26	22, 25	anbi12d 634	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))))
27	11, 16, 21, 26	rexprgf 4609	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ ((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ∨ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))))
28		dfsbcq 3696	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
29		eqeq2 2749	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐴))
30	28, 29	imbi12d 348	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
31		dfsbcq 3696	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → ([𝑐 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
32		eqeq2 2749	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐵))
33	31, 32	imbi12d 348	. . . . . . 7 ⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)))
34	30, 33	ralprg 4610	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ (([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵))))
35		eqidd 2738	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
36	35	biantrur 534	. . . . . . 7 ⊢ (([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)))
37	22	sbcieg 3734	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑥]𝜑 ↔ 𝜒))
38	37	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐵 / 𝑥]𝜑 ↔ 𝜒))
39	38	imbi1d 345	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (𝜒 → 𝐴 = 𝐵)))
40	36, 39	bitr3id 288	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)) ↔ (𝜒 → 𝐴 = 𝐵)))
41	34, 40	bitrd 282	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ (𝜒 → 𝐴 = 𝐵)))
42	41	anbi2d 632	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ (𝜓 ∧ (𝜒 → 𝐴 = 𝐵))))
43		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐴))
44	28, 43	imbi12d 348	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴)))
45		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐵))
46	31, 45	imbi12d 348	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)))
47	44, 46	ralprg 4610	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵))))
48		eqidd 2738	. . . . . . . . 9 ⊢ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)
49	48	biantru 533	. . . . . . . 8 ⊢ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)))
50	17	sbcieg 3734	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
51	50	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
52	51	imbi1d 345	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐵 = 𝐴)))
53	49, 52	bitr3id 288	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)) ↔ (𝜓 → 𝐵 = 𝐴)))
54	47, 53	bitrd 282	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ (𝜓 → 𝐵 = 𝐴)))
55	54	anbi2d 632	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐵 = 𝐴))))
56		eqcom 2744	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
57	56	imbi2i 339	. . . . . 6 ⊢ ((𝜓 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐴 = 𝐵))
58	57	anbi2i 626	. . . . 5 ⊢ ((𝜒 ∧ (𝜓 → 𝐵 = 𝐴)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))
59	55, 58	bitrdi 290	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))))
60	42, 59	orbi12d 919	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ∨ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))) ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))
61	27, 60	bitrd 282	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))
62	5, 61	syl5bb 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  ∃!wreu 3063  [wsbc 3694  {cpr 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-v 3410  df-sbc 3695  df-un 3871  df-sn 4542  df-pr 4544

This theorem is referenced by:  reuprg  4619