Theorem reusngf 4605

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 3731	. . 3 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜑
2		nfsbc1v 3731	. . 3 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
3		sbceq1a 3722	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
4		dfsbcq 3713	. . 3 ⊢ (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑 ↔ [𝑐 / 𝑥]𝜑))
5	1, 2, 3, 4	reu8nf 3806	. 2 ⊢ (∃!𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)))
6		rexsngf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
7		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
8		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑐
9	1, 8	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
10	7, 9	nfralw 3149	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)
11	6, 10	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))
12		rexsngf.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13		eqeq1 2742	. . . . . . 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 3120	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))
16	12, 15	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))))
17	11, 16	rexsngf 4603	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))))
18		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑐([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
19		dfsbcq 3713	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
20		eqeq2 2750	. . . . . . 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 4604	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)))
23	22	anbi2d 628	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴))))
24		eqidd 2739	. . . . 5 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴)
25	24	biantru 529	. . . 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	syl5bb 282	1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  [wsbc 3711  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-v 3424  df-sbc 3712  df-sn 4559

This theorem is referenced by:  reusng  4608  rmosn  4652