Theorem ssralv2 44881

Description: Quantification restricted to a subclass for two quantifiers. ssralv 4004 for two quantifiers. The proof of ssralv2 44881 was automatically generated by minimizing the automatically translated proof of ssralv2VD 45215. 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.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑦,𝐶   𝑥,𝐷   𝑦,𝐷

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem ssralv2

Step	Hyp	Ref	Expression
1		nfv 1916	. 2 ⊢ Ⅎ𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
2		nfra1 3262	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑
3		ssralv 4004	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑))
4	3	adantr 480	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑))
5		df-ral 3053	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
6	4, 5	imbitrdi 251	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)))
7		sp 2191	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
8	6, 7	syl6 35	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)))
9		ssralv 4004	. . . 4 ⊢ (𝐶 ⊆ 𝐷 → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑))
10	9	adantl 481	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑))
11	8, 10	syl6d 75	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑)))
12	1, 2, 11	ralrimd 3243	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-ral 3053  df-ss 3920

This theorem is referenced by:  ordelordALT  44887  ordelordALTVD  45216