Theorem cbvral8vw

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

	Ref	Expression
Hypotheses	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	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 )

Proof

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