# 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 ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ∀ 𝑥𝐴𝑦𝐶 𝜑 ) )

### Proof

Step Hyp Ref Expression
1 nfv 𝑥 ( 𝐴𝐵𝐶𝐷 )
2 nfra1 𝑥𝑥𝐵𝑦𝐷 𝜑
3 ssralv ( 𝐴𝐵 → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ∀ 𝑥𝐴𝑦𝐷 𝜑 ) )
4 3 adantr ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ∀ 𝑥𝐴𝑦𝐷 𝜑 ) )
5 df-ral ( ∀ 𝑥𝐴𝑦𝐷 𝜑 ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦𝐷 𝜑 ) )
6 4 5 syl6ib ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦𝐷 𝜑 ) ) )
7 sp ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦𝐷 𝜑 ) → ( 𝑥𝐴 → ∀ 𝑦𝐷 𝜑 ) )
8 6 7 syl6 ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ( 𝑥𝐴 → ∀ 𝑦𝐷 𝜑 ) ) )
9 ssralv ( 𝐶𝐷 → ( ∀ 𝑦𝐷 𝜑 → ∀ 𝑦𝐶 𝜑 ) )
10 9 adantl ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑦𝐷 𝜑 → ∀ 𝑦𝐶 𝜑 ) )
11 8 10 syl6d ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ( 𝑥𝐴 → ∀ 𝑦𝐶 𝜑 ) ) )
12 1 2 11 ralrimd ( ( 𝐴𝐵𝐶𝐷 ) → ( ∀ 𝑥𝐵𝑦𝐷 𝜑 → ∀ 𝑥𝐴𝑦𝐶 𝜑 ) )