Theorem reu8nf 3820

Description: Restricted uniqueness using implicit substitution. This version of reu8 3678 uses a nonfreeness 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 1916	. . 3 ⊢ Ⅎ𝑤𝜑
2		reu8nf.2	. . 3 ⊢ Ⅎ𝑥𝜒
3		reu8nf.3	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
4	1, 2, 3	cbvreuw 3379	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑤 ∈ 𝐴 𝜒)
5		reu8nf.4	. . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))
6	5	reu8 3678	. 2 ⊢ (∃!𝑤 ∈ 𝐴 𝜒 ↔ ∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)))
7		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝐴
8		reu8nf.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
9		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 𝑦
10	8, 9	nfim 1898	. . . . 5 ⊢ Ⅎ𝑥(𝜓 → 𝑤 = 𝑦)
11	7, 10	nfralw 3290	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)
12	2, 11	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦))
13		nfv 1916	. . 3 ⊢ Ⅎ𝑤(𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
14	3	bicomd 222	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜒 ↔ 𝜑))
15	14	equcoms 2022	. . . 4 ⊢ (𝑤 = 𝑥 → (𝜒 ↔ 𝜑))
16		equequ1 2027	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
17	16	imbi2d 340	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝜓 → 𝑤 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
18	17	ralbidv 3170	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
19	15, 18	anbi12d 631	. . 3 ⊢ (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
20	12, 13, 19	cbvrexw 3286	. 2 ⊢ (∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
21	4, 6, 20	3bitri 296	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1784  ∀wral 3061  ∃wrex 3070  ∃!wreu 3347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-10 2136  ax-11 2153  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350

This theorem is referenced by:  reusngf  4619  reuprg0  4649  reuop  6225  reuccatpfxs1  14550  reupr  45314