Theorem reuprg0 4641

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 3795	. . 3 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜑
2		nfsbc1v 3795	. . 3 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
3		sbceq1a 3786	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
4		dfsbcq 3777	. . 3 ⊢ (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑 ↔ [𝑐 / 𝑥]𝜑))
5	1, 2, 3, 4	reu8nf 3863	. 2 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)))
6		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥𝜓
7		nfcv 2980	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑐
9	1, 8	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
10	7, 9	nfralw 3228	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
11	6, 10	nfan 1899	. . . 4 ⊢ Ⅎ𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))
12		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥𝜒
13		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝐵 = 𝑐
14	1, 13	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)
15	7, 14	nfralw 3228	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)
16	12, 15	nfan 1899	. . . 4 ⊢ Ⅎ𝑥(𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))
17		reuprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
18		eqeq1 2828	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑐 ↔ 𝐴 = 𝑐))
19	18	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
20	19	ralbidv 3200	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
21	17, 20	anbi12d 632	. . . 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 2828	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑐 ↔ 𝐵 = 𝑐))
24	23	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))
25	24	ralbidv 3200	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))
26	22, 25	anbi12d 632	. . . 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 4634	. . 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 3777	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
29		eqeq2 2836	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐴))
30	28, 29	imbi12d 347	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
31		dfsbcq 3777	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → ([𝑐 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
32		eqeq2 2836	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐵))
33	31, 32	imbi12d 347	. . . . . . 7 ⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)))
34	30, 33	ralprg 4635	. . . . . 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 2825	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
36	35	biantrur 533	. . . . . . 7 ⊢ (([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)))
37	22	sbcieg 3813	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑥]𝜑 ↔ 𝜒))
38	37	adantl 484	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐵 / 𝑥]𝜑 ↔ 𝜒))
39	38	imbi1d 344	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (𝜒 → 𝐴 = 𝐵)))
40	36, 39	syl5bbr 287	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)) ↔ (𝜒 → 𝐴 = 𝐵)))
41	34, 40	bitrd 281	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ (𝜒 → 𝐴 = 𝐵)))
42	41	anbi2d 630	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ (𝜓 ∧ (𝜒 → 𝐴 = 𝐵))))
43		eqeq2 2836	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐴))
44	28, 43	imbi12d 347	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴)))
45		eqeq2 2836	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐵))
46	31, 45	imbi12d 347	. . . . . . . 8 ⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)))
47	44, 46	ralprg 4635	. . . . . . 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 2825	. . . . . . . . 9 ⊢ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)
49	48	biantru 532	. . . . . . . 8 ⊢ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)))
50	17	sbcieg 3813	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
51	50	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
52	51	imbi1d 344	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐵 = 𝐴)))
53	49, 52	syl5bbr 287	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)) ↔ (𝜓 → 𝐵 = 𝐴)))
54	47, 53	bitrd 281	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ (𝜓 → 𝐵 = 𝐴)))
55	54	anbi2d 630	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐵 = 𝐴))))
56		eqcom 2831	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
57	56	imbi2i 338	. . . . . 6 ⊢ ((𝜓 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐴 = 𝐵))
58	57	anbi2i 624	. . . . 5 ⊢ ((𝜒 ∧ (𝜓 → 𝐵 = 𝐴)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))
59	55, 58	syl6bb 289	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))))
60	42, 59	orbi12d 915	. . 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 281	. 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 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  ∃!wreu 3143  [wsbc 3775  {cpr 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-reu 3148  df-v 3499  df-sbc 3776  df-un 3944  df-sn 4571  df-pr 4573

This theorem is referenced by:  reuprg  4642