Theorem 6ralbidv 3222

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 5-Mar-2025.)

Hypothesis

Ref	Expression
6ralbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
6ralbidv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜒))

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

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

Proof of Theorem 6ralbidv

Step	Hyp	Ref	Expression
1		6ralbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	2ralbidv 3217	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜓 ↔ ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜒))
3	2	4ralbidv 3221	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ral 3061

This theorem is referenced by:  mulsprop  27499