Theorem r2alf 3292

Description: Double restricted universal quantification. For a version based on fewer axioms see r2al 3207. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3159. (Revised by Wolf Lammen, 9-Jan-2020.)

Hypothesis

Ref	Expression
r2alf.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
r2alf	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem r2alf

Step	Hyp	Ref	Expression
1		r2alf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2923	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3	2	19.21 2249	. 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
4	3	r2allem 3159	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   ∈ wcel 2149  Ⅎwnfc 2916  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-clel 2844  df-nfc 2918  df-ral 3086

This theorem is referenced by:  r2exf  3293  ralcomf  3309