Theorem cbvral8vw 3242

Description: Change bound variables of octuple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 2-Mar-2025.)

Hypotheses

Ref	Expression
cbvral8vw.1	⊢ (𝑥 = 𝑎 → (𝜑 ↔ 𝜒))
cbvral8vw.2	⊢ (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))
cbvral8vw.3	⊢ (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))
cbvral8vw.4	⊢ (𝑤 = 𝑑 → (𝜏 ↔ 𝜂))
cbvral8vw.5	⊢ (𝑝 = 𝑒 → (𝜂 ↔ 𝜁))
cbvral8vw.6	⊢ (𝑞 = 𝑓 → (𝜁 ↔ 𝜎))
cbvral8vw.7	⊢ (𝑟 = 𝑔 → (𝜎 ↔ 𝜌))
cbvral8vw.8	⊢ (𝑠 = ℎ → (𝜌 ↔ 𝜓))

Assertion

Ref	Expression
cbvral8vw	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐺 ∀ℎ ∈ 𝐻 𝜓)

Distinct variable groups:   𝑥,𝑎,𝐴   𝑥,𝑦,𝐵,𝑎   𝑦,𝑏,𝐵   𝑥,𝐶,𝑎,𝑦   𝐶,𝑏   𝑧,𝑐,𝐶   𝑥,𝑧,𝐷,𝑦,𝑎   𝐷,𝑏   𝐷,𝑐   𝑤,𝑑,𝐷   𝑥,𝑤,𝐸,𝑦,𝑧,𝑎   𝐸,𝑏   𝐸,𝑐   𝐸,𝑑   𝑒,𝑝,𝐸   𝑥,𝑝,𝐹,𝑦,𝑧,𝑤,𝑎   𝐹,𝑏   𝐹,𝑐   𝐹,𝑑   𝑒,𝐹   𝑓,𝑞,𝐹   𝑥,𝑞,𝐺,𝑦,𝑧,𝑤,𝑝,𝑎   𝐺,𝑏   𝐺,𝑐   𝐺,𝑑   𝑒,𝐺   𝑓,𝐺   𝑔,𝑟,𝐺   𝑥,𝑟,𝐻,𝑦,𝑧,𝑤,𝑝,𝑞,𝑎   𝐻,𝑏   𝐻,𝑐   𝐻,𝑑   𝑒,𝐻   𝑓,𝐻   𝑔,𝐻   ℎ,𝑠,𝐻   𝜑,𝑎   𝑠,𝑎   𝜒,𝑏   𝑝,𝑏,𝑠,𝑞,𝑟,𝑤,𝑧   𝑝,𝑐,𝑞,𝑟,𝑠   𝜃,𝑐   𝑤,𝑐   𝜒,𝑥   𝑝,𝑑,𝑞,𝑟,𝑠   𝜏,𝑑   𝜂,𝑒   𝑒,𝑞,𝑟,𝑠   𝜂,𝑤   𝑓,𝑟,𝑠   𝜁,𝑓   𝑔,𝑠   𝜎,𝑔   𝜌,ℎ   𝜁,𝑝   𝜓,𝑠   𝜎,𝑞   𝜌,𝑟   𝑥,𝑠,𝑦   𝜏,𝑧   𝜃,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜒(𝑦,𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑒,𝑔,ℎ,𝑠,𝑟,𝑞,𝑎,𝑏,𝑐,𝑑)   𝜎(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,ℎ,𝑠,𝑟,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜌(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑔,𝑠,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑦,𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑏,𝑐,𝑑)   𝐵(𝑧,𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑐,𝑑)   𝐶(𝑤,𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝,𝑑)   𝐷(𝑒,𝑓,𝑔,ℎ,𝑠,𝑟,𝑞,𝑝)   𝐸(𝑓,𝑔,ℎ,𝑠,𝑟,𝑞)   𝐹(𝑔,ℎ,𝑠,𝑟)   𝐺(ℎ,𝑠)

Proof of Theorem cbvral8vw

Step	Hyp	Ref	Expression
1		cbvral8vw.1	. . . 4 ⊢ (𝑥 = 𝑎 → (𝜑 ↔ 𝜒))
2	1	4ralbidv 3221	. . 3 ⊢ (𝑥 = 𝑎 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜑 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜒))
3		cbvral8vw.2	. . . 4 ⊢ (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))
4	3	4ralbidv 3221	. . 3 ⊢ (𝑦 = 𝑏 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜒 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜃))
5		cbvral8vw.3	. . . 4 ⊢ (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))
6	5	4ralbidv 3221	. . 3 ⊢ (𝑧 = 𝑐 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜃 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜏))
7		cbvral8vw.4	. . . 4 ⊢ (𝑤 = 𝑑 → (𝜏 ↔ 𝜂))
8	7	4ralbidv 3221	. . 3 ⊢ (𝑤 = 𝑑 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜏 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜂))
9	2, 4, 6, 8	cbvral4vw 3240	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜂)
10		cbvral8vw.5	. . . 4 ⊢ (𝑝 = 𝑒 → (𝜂 ↔ 𝜁))
11		cbvral8vw.6	. . . 4 ⊢ (𝑞 = 𝑓 → (𝜁 ↔ 𝜎))
12		cbvral8vw.7	. . . 4 ⊢ (𝑟 = 𝑔 → (𝜎 ↔ 𝜌))
13		cbvral8vw.8	. . . 4 ⊢ (𝑠 = ℎ → (𝜌 ↔ 𝜓))
14	10, 11, 12, 13	cbvral4vw 3240	. . 3 ⊢ (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜂 ↔ ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐺 ∀ℎ ∈ 𝐻 𝜓)
15	14	4ralbii 3130	. 2 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜂 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐺 ∀ℎ ∈ 𝐻 𝜓)
16	9, 15	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐺 ∀ℎ ∈ 𝐻 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clel 2809  df-ral 3061

This theorem is referenced by: (None)