Metamath Proof Explorer

Theorem ssralv2

Description: Quantification restricted to a subclass for two quantifiers. ssralv for two quantifiers. The proof of ssralv2 was automatically generated by minimizing the automatically translated proof of ssralv2VD . The automatic translation is by the tools program translate__without__overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion ssralv2 ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → ∀ x ∈ A ∀ y ∈ C φ

Proof

Step Hyp Ref Expression
1 nfv ⊢ Ⅎ x A ⊆ B ∧ C ⊆ D
2 nfra1 ⊢ Ⅎ x ∀ x ∈ B ∀ y ∈ D φ
3 ssralv ⊢ A ⊆ B → ∀ x ∈ B ∀ y ∈ D φ → ∀ x ∈ A ∀ y ∈ D φ
4 3 adantr ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → ∀ x ∈ A ∀ y ∈ D φ
5 df-ral ⊢ ∀ x ∈ A ∀ y ∈ D φ ↔ ∀ x x ∈ A → ∀ y ∈ D φ
6 4 5 imbitrdi ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → ∀ x x ∈ A → ∀ y ∈ D φ
7 sp ⊢ ∀ x x ∈ A → ∀ y ∈ D φ → x ∈ A → ∀ y ∈ D φ
8 6 7 syl6 ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → x ∈ A → ∀ y ∈ D φ
9 ssralv ⊢ C ⊆ D → ∀ y ∈ D φ → ∀ y ∈ C φ
10 9 adantl ⊢ A ⊆ B ∧ C ⊆ D → ∀ y ∈ D φ → ∀ y ∈ C φ
11 8 10 syl6d ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → x ∈ A → ∀ y ∈ C φ
12 1 2 11 ralrimd ⊢ A ⊆ B ∧ C ⊆ D → ∀ x ∈ B ∀ y ∈ D φ → ∀ x ∈ A ∀ y ∈ C φ