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Metamath Proof Explorer

Theorem rspc6v

Description: 6-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025)

Ref Expression
Hypotheses rspc6v.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜒 ) )
rspc6v.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
rspc6v.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) )
rspc6v.4 ⊢ ( 𝑤 = 𝐷 → ( 𝜏 ↔ 𝜂 ) )
rspc6v.5 ⊢ ( 𝑝 = 𝐸 → ( 𝜂 ↔ 𝜁 ) )
rspc6v.6 ⊢ ( 𝑞 = 𝐹 → ( 𝜁 ↔ 𝜓 ) )
Assertion rspc6v ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ∧ ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 rspc6v.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜒 ) )
2 rspc6v.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
3 rspc6v.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) )
4 rspc6v.4 ⊢ ( 𝑤 = 𝐷 → ( 𝜏 ↔ 𝜂 ) )
5 rspc6v.5 ⊢ ( 𝑝 = 𝐸 → ( 𝜂 ↔ 𝜁 ) )
6 rspc6v.6 ⊢ ( 𝑞 = 𝐹 → ( 𝜁 ↔ 𝜓 ) )
7 1 2ralbidv ⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜑 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜒 ) )
8 2 2ralbidv ⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜒 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜃 ) )
9 3 2ralbidv ⊢ ( 𝑧 = 𝐶 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜃 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜏 ) )
10 4 2ralbidv ⊢ ( 𝑤 = 𝐷 → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜏 ↔ ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜂 ) )
11 7 8 9 10 rspc4v ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜑 → ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜂 ) )
12 5 6 rspc2v ⊢ ( ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) → ( ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜂 → 𝜓 ) )
13 11 12 syl9 ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ) → ( ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜑 → 𝜓 ) ) )
14 13 3impia ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) ∧ ( 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑇 ∀ 𝑤 ∈ 𝑈 ∀ 𝑝 ∈ 𝑉 ∀ 𝑞 ∈ 𝑊 𝜑 → 𝜓 ) )