This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025)

Ref Expression
Hypothesis 3ralbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion 3ralbidv ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜒 ) )

Proof

Step Hyp Ref Expression
1 3ralbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 1 ralbidv ⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑧 ∈ 𝐶 𝜒 ) )
3 2 2ralbidv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜒 ) )