Theorem rexreusng 4612

Description: Restricted existential uniqueness over a singleton is equivalent to a restricted existential quantification over a singleton. (Contributed by AV, 3-Apr-2023.)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem rexreusng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2739	. . . . 5 ⊢ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
2		nfsbc1v 3731	. . . . . . . 8 ⊢ Ⅎ𝑦[𝐴 / 𝑦][𝐴 / 𝑥]𝜑
3		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑦[𝐴 / 𝑥]𝜑
4	2, 3	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑦([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑)
5		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = 𝐴
6	4, 5	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑦(([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
7		sbceq1a 3722	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝐴 / 𝑥]𝜑))
8		dfsbcq2 3714	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
9	7, 8	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑)))
10		eqeq2 2750	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
11	9, 10	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦) ↔ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)))
12	6, 11	ralsngf 4604	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦) ↔ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)))
13	1, 12	mpbiri 257	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦))
14		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
15		nfsbc1v 3731	. . . . . . . 8 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
16		nfs1v 2155	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
17	15, 16	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑥([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)
18		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑦
19	17, 18	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑥(([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
20	14, 19	nfralw 3149	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
21		sbceq1a 3722	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
22	21	anbi1d 629	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)))
23		eqeq1 2742	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
24	22, 23	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
25	24	ralbidv 3120	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
26	20, 25	ralsngf 4604	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
27	13, 26	mpbird 256	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
28	27	biantrud 531	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
29		reu2 3655	. 2 ⊢ (∃!𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
30	28, 29	bitr4di 288	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  [wsb 2068   ∈ 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: (None)