Theorem reu8nf 3860

Description: Restricted uniqueness using implicit substitution. This version of reu8 3724 uses a non-freeness hypothesis for 𝑥 and 𝜓 instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)

Hypotheses

Ref	Expression
reu8nf.1	⊢ Ⅎ𝑥𝜓
reu8nf.2	⊢ Ⅎ𝑥𝜒
reu8nf.3	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
reu8nf.4	⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
reu8nf	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑤,𝑦,𝐴   𝜑,𝑤   𝜓,𝑤   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑤)

Proof of Theorem reu8nf

Step	Hyp	Ref	Expression
1		nfv 1911	. . 3 ⊢ Ⅎ𝑤𝜑
2		reu8nf.2	. . 3 ⊢ Ⅎ𝑥𝜒
3		reu8nf.3	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
4	1, 2, 3	cbvreuw 3444	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑤 ∈ 𝐴 𝜒)
5		reu8nf.4	. . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))
6	5	reu8 3724	. 2 ⊢ (∃!𝑤 ∈ 𝐴 𝜒 ↔ ∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)))
7		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥𝐴
8		reu8nf.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
9		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 𝑦
10	8, 9	nfim 1893	. . . . 5 ⊢ Ⅎ𝑥(𝜓 → 𝑤 = 𝑦)
11	7, 10	nfralw 3225	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)
12	2, 11	nfan 1896	. . 3 ⊢ Ⅎ𝑥(𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦))
13		nfv 1911	. . 3 ⊢ Ⅎ𝑤(𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
14	3	bicomd 225	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜒 ↔ 𝜑))
15	14	equcoms 2023	. . . 4 ⊢ (𝑤 = 𝑥 → (𝜒 ↔ 𝜑))
16		equequ1 2028	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
17	16	imbi2d 343	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝜓 → 𝑤 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
18	17	ralbidv 3197	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
19	15, 18	anbi12d 632	. . 3 ⊢ (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
20	12, 13, 19	cbvrexw 3443	. 2 ⊢ (∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
21	4, 6, 20	3bitri 299	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎwnf 1780  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145

This theorem is referenced by:  reusngf  4606  reuprg0  4632  reuop  6139  reuccatpfxs1  14103  reupr  43677