Theorem rexreusng 4641

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 2764	. . . . 5 ⊢ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
2		nfsbc1v 3762	. . . . . . . 8 ⊢ Ⅎ𝑦[𝐴 / 𝑦][𝐴 / 𝑥]𝜑
3		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑦[𝐴 / 𝑥]𝜑
4	2, 3	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑦([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑)
5		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = 𝐴
6	4, 5	nfim 1931	. . . . . 6 ⊢ Ⅎ𝑦(([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
7		sbceq1a 3753	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝐴 / 𝑥]𝜑))
8		dfsbcq2 3745	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
9	7, 8	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑦][𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜑)))
10		eqeq2 2775	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
11	9, 10	imbi12d 347	. . . . . 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 4635	. . . . 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 261	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦))
14		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
15		nfsbc1v 3762	. . . . . . . 8 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
16		nfs1v 2196	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
17	15, 16	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑥([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)
18		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝑦
19	17, 18	nfim 1931	. . . . . 6 ⊢ Ⅎ𝑥(([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
20	14, 19	nfralw 3313	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
21		sbceq1a 3753	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
22	21	anbi1d 643	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)))
23		eqeq1 2767	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
24	22, 23	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
25	24	ralbidv 3189	. . . . 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 4635	. . . 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 260	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
28	27	biantrud 541	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
29		reu2 3686	. 2 ⊢ (∃!𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
30	28, 29	bitr4di 292	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  [wsb 2101   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367  [wsbc 3742  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-reu 3370  df-v 3456  df-sbc 3743  df-sn 4589

This theorem is used by: (None)