Theorem ss2ralv 4079

Description: Two quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)

Assertion

Ref	Expression
ss2ralv	⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ss2ralv

Step	Hyp	Ref	Expression
1		ssralv 4077	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐴 𝜑))
2	1	ralimdv 3175	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑))
3		ssralv 4077	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))
4	2, 3	syld 47	1 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4  ∀wral 3067   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3068  df-ss 3993

This theorem is referenced by:  poss  5609  soss  5628  dffi3  9500  isercolllem1  15713  cfilres  25349  lgsdchr  27417  dffltz  42589