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Metamath Proof Explorer

Theorem rspc3dv

Description: 3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025)

Ref Expression
Hypotheses rspc3dv.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
rspc3dv.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜃 ↔ 𝜏 ) )
rspc3dv.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜏 ↔ 𝜒 ) )
rspc3dv.4 ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 )
rspc3dv.5 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
rspc3dv.6 ⊢ ( 𝜑 → 𝐵 ∈ 𝐸 )
rspc3dv.7 ⊢ ( 𝜑 → 𝐶 ∈ 𝐹 )
Assertion rspc3dv ( 𝜑 → 𝜒 )

Proof

Step Hyp Ref Expression
1 rspc3dv.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
2 rspc3dv.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜃 ↔ 𝜏 ) )
3 rspc3dv.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜏 ↔ 𝜒 ) )
4 rspc3dv.4 ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 )
5 rspc3dv.5 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
6 rspc3dv.6 ⊢ ( 𝜑 → 𝐵 ∈ 𝐸 )
7 rspc3dv.7 ⊢ ( 𝜑 → 𝐶 ∈ 𝐹 )
8 5 6 7 3jca ⊢ ( 𝜑 → ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )
9 1 2 3 rspc3v ⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 → 𝜒 ) )
10 8 4 9 sylc ⊢ ( 𝜑 → 𝜒 )