Theorem 4ralbii 3142

Description: Inference adding four restricted universal quantifiers to both sides of an equivalence. (Contributed by Scott Fenton, 28-Feb-2025.)

Hypothesis

Ref	Expression
4ralbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem 4ralbii

Step	Hyp	Ref	Expression
1		4ralbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	ralbii 3110	. 2 ⊢ (∀𝑤 ∈ 𝐷 𝜑 ↔ ∀𝑤 ∈ 𝐷 𝜓)
3	2	3ralbii 3141	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-ral 3079

This theorem is used by:  cbvral6vw  3250  cbvral8vw  3251