Theorem reu8nf 3087

Description: Restricted uniqueness using implicit substitution. This version of reu8 2976 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 1552	. . 3 ⊢ Ⅎ𝑤𝜑
2		reu8nf.2	. . 3 ⊢ Ⅎ𝑥𝜒
3		reu8nf.3	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
4	1, 2, 3	cbvreuw 2737	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑤 ∈ 𝐴 𝜒)
5		reu8nf.4	. . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))
6	5	reu8 2976	. 2 ⊢ (∃!𝑤 ∈ 𝐴 𝜒 ↔ ∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)))
7		nfcv 2350	. . . . 5 ⊢ Ⅎ𝑥𝐴
8		reu8nf.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
9		nfv 1552	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 𝑦
10	8, 9	nfim 1596	. . . . 5 ⊢ Ⅎ𝑥(𝜓 → 𝑤 = 𝑦)
11	7, 10	nfralw 2545	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)
12	2, 11	nfan 1589	. . 3 ⊢ Ⅎ𝑥(𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦))
13		nfv 1552	. . 3 ⊢ Ⅎ𝑤(𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
14	3	bicomd 141	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜒 ↔ 𝜑))
15	14	equcoms 1732	. . . 4 ⊢ (𝑤 = 𝑥 → (𝜒 ↔ 𝜑))
16		equequ1 1736	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
17	16	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝜓 → 𝑤 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
18	17	ralbidv 2508	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
19	15, 18	anbi12d 473	. . 3 ⊢ (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
20	12, 13, 19	cbvrexw 2736	. 2 ⊢ (∃𝑤 ∈ 𝐴 (𝜒 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑤 = 𝑦)) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
21	4, 6, 20	3bitri 206	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  Ⅎwnf 1484  ∀wral 2486  ∃wrex 2487  ∃!wreu 2488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-v 2778

This theorem is referenced by:  reuccatpfxs1  11238