Theorem 3ralbii 35504

Description: Inference adding three restricted universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019.)

Hypothesis

Ref	Expression
3ralbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
3ralbii	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)

Proof of Theorem 3ralbii

Step	Hyp	Ref	Expression
1		3ralbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	2ralbii 3166	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
3	2	ralbii 3165	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)

Syntax hints:   ↔ wb 208  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ral 3143

This theorem is referenced by: (None)