Theorem ss2ralv 4009

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 4007	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐴 𝜑))
2	1	ralimdv 3181	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑))
3		ssralv 4007	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))
4	2, 3	syld 48	1 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wral 3081   ⊆ wss 3906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3082  df-ss 3923

This theorem is used by:  poss  5573  soss  5591  dffi3  9394  isercolllem1  15736  cfilres  25486  lgsdchr  27550  dffltz  43399