Theorem r2alan 36457

Description: Double restricted universal quantification, special case. (Contributed by Peter Mazsa, 17-Jun-2020.)

Assertion

Ref	Expression
r2alan	⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

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

Proof of Theorem r2alan

Step	Hyp	Ref	Expression
1		impexp 452	. . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)))
2	1	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)))
3		r2al 3187	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)))
4	2, 3	bitr4i 278	1 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1537   ∈ wcel 2104  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1780  df-ral 3062

This theorem is referenced by:  antisymrelres  36977