Theorem 2ralbida 3162

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)

Hypotheses

Ref	Expression
2ralbida.1	⊢ Ⅎ𝑥𝜑
2ralbida.2	⊢ Ⅎ𝑦𝜑
2ralbida.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
2ralbida	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2ralbida

Step	Hyp	Ref	Expression
1		2ralbida.1	. 2 ⊢ Ⅎ𝑥𝜑
2		2ralbida.2	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfv 1917	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
4	2, 3	nfan 1902	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴)
5		2ralbida.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))
6	5	anassrs 468	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒))
7	4, 6	ralbida 3159	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒))
8	1, 7	ralbida 3159	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-ral 3069

This theorem is referenced by: (None)