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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 ) )