Theorem 4ralbidv 3230

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

Hypothesis

Ref	Expression
4ralbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
4ralbidv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧   𝜑,𝑤

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑧, 𝑤)   𝜒(𝑥, 𝑦, 𝑧, 𝑤)   𝐴(𝑥, 𝑦, 𝑧, 𝑤)   𝐵(𝑥, 𝑦, 𝑧, 𝑤)   𝐶(𝑥, 𝑦, 𝑧, 𝑤)   𝐷(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem 4ralbidv

Step	Hyp	Ref	Expression
1		4ralbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	ralbidv 3185	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐷 𝜓 ↔ ∀𝑤 ∈ 𝐷 𝜒))
3	2	3ralbidv 3229	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3077

This theorem is used by:  6ralbidv  3231  cbvral8vw  3249  rspc8v  3597  nelsubc3lem  50100