Theorem ssralv2 44505

Description: Quantification restricted to a subclass for two quantifiers. ssralv 4006 for two quantifiers. The proof of ssralv2 44505 was automatically generated by minimizing the automatically translated proof of ssralv2VD 44839. 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 1914	. 2 ⊢ Ⅎ𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
2		nfra1 3253	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑
3		ssralv 4006	. . . . . 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 3045	. . . . 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 2184	. . . 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 4006	. . . 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 3234	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-ral 3045  df-ss 3922

This theorem is referenced by:  ordelordALT  44511  ordelordALTVD  44840