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Metamath Proof Explorer

Theorem rspc8v

Description: 8-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025)

		Ref	Expression
	Hypotheses	rspc8v.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜒 ) )
		rspc8v.2	⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
		rspc8v.3	⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) )
		rspc8v.4	⊢ ( 𝑤 = 𝐷 → ( 𝜏 ↔ 𝜂 ) )
		rspc8v.5	⊢ ( 𝑝 = 𝐸 → ( 𝜂 ↔ 𝜁 ) )
		rspc8v.6	⊢ ( 𝑞 = 𝐹 → ( 𝜁 ↔ 𝜎 ) )
		rspc8v.7	⊢ ( 𝑟 = 𝐺 → ( 𝜎 ↔ 𝜌 ) )
		rspc8v.8	⊢ ( 𝑠 = 𝐻 → ( 𝜌 ↔ 𝜓 ) )
	Assertion	rspc8v	⊢ ( ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ) ∧ ( ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rspc8v.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜒 ) )
2		rspc8v.2	⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
3		rspc8v.3	⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) )
4		rspc8v.4	⊢ ( 𝑤 = 𝐷 → ( 𝜏 ↔ 𝜂 ) )
5		rspc8v.5	⊢ ( 𝑝 = 𝐸 → ( 𝜂 ↔ 𝜁 ) )
6		rspc8v.6	⊢ ( 𝑞 = 𝐹 → ( 𝜁 ↔ 𝜎 ) )
7		rspc8v.7	⊢ ( 𝑟 = 𝐺 → ( 𝜎 ↔ 𝜌 ) )
8		rspc8v.8	⊢ ( 𝑠 = 𝐻 → ( 𝜌 ↔ 𝜓 ) )
9	1	4ralbidv	⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜑 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜒 ) )
10	2	4ralbidv	⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜒 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜃 ) )
11	3	4ralbidv	⊢ ( 𝑧 = 𝐶 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜃 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜏 ) )
12	4	4ralbidv	⊢ ( 𝑤 = 𝐷 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜏 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜂 ) )
13	9 10 11 12	rspc4v	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜑 → ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜂 ) )
14	5 6 7 8	rspc4v	⊢ ( ( ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) ) → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜂 → 𝜓 ) )
15	13 14	sylan9	⊢ ( ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ) ∧ ( ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 ∀ 𝑟 ∈ 𝑋 ∀ 𝑠 ∈ 𝑌 𝜑 → 𝜓 ) )