Theorem r2alf 3270

Description: Double restricted universal quantification. For a version based on fewer axioms see r2al 3186. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3134. (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 2882	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3	2	19.21 2192	. 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
4	3	r2allem 3134	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-clel 2802  df-nfc 2877  df-ral 3054

This theorem is referenced by:  r2exf  3271  ralcomf  3291