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∈Dφ→∀x∈A∀y∈Cφ

Proof

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