Theorem reusngf 4676

Description: Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)

Hypotheses

Ref	Expression
rexsngf.1	⊢ Ⅎ𝑥𝜓
rexsngf.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
reusngf	⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem reusngf

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

Step	Hyp	Ref	Expression
1		nfsbc1v 3797	. . 3 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜑
2		nfsbc1v 3797	. . 3 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
3		sbceq1a 3788	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
4		dfsbcq 3779	. . 3 ⊢ (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑 ↔ [𝑐 / 𝑥]𝜑))
5	1, 2, 3, 4	reu8nf 3871	. 2 ⊢ (∃!𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)))
6		rexsngf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
7		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
8		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑐
9	1, 8	nfim 1899	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
10	7, 9	nfralw 3308	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
11	6, 10	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))
12		rexsngf.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13		eqeq1 2736	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑐 ↔ 𝐴 = 𝑐))
14	13	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
15	14	ralbidv 3177	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
16	12, 15	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))))
17	11, 16	rexsngf 4674	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))))
18		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑐([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
19		dfsbcq 3779	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
20		eqeq2 2744	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐴))
21	19, 20	imbi12d 344	. . . . . 6 ⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
22	18, 21	ralsngf 4675	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
23	22	anbi2d 629	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴))))
24		eqidd 2733	. . . . 5 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
25	24	biantru 530	. . . 4 ⊢ (𝜓 ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
26	23, 25	bitr4di 288	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ 𝜓))
27	17, 26	bitrd 278	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ 𝜓))
28	5, 27	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  ∃!wreu 3374  [wsbc 3777  {csn 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-v 3476  df-sbc 3778  df-sn 4629

This theorem is referenced by:  reusng  4679  rmosn  4723