Theorem ssralv2VD 44985

Description: Quantification restricted to a subclass for two quantifiers. ssralv 3999 for two quantifiers. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ssralv2 44651 is ssralv2VD 44985 without virtual deductions and was automatically derived from ssralv2VD 44985.

1::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   )
2::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑   )
3:1:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐴 ⊆ 𝐵   )
4:3,2:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐷𝜑   )
5:4:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑)   )
6:5:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑)   )
7::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8:7,6:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐷𝜑   )
9:1:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐶 ⊆ 𝐷   )
10:9,8:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐶𝜑   )
11:10:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑)   )
12::	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
13::	⊢ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑)
14:12,13,11:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑)   )
15:14:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑   )
16:15:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)    ▶   (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑)   )
qed:16:	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑))

(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ssralv2VD	⊢ ((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 ssralv2VD

Step	Hyp	Ref	Expression
1		ax-5 1911	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
2		hbra1 3270	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑)
3		idn1 44694	. . . . . . . 8 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   )
4		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → 𝐶 ⊆ 𝐷)
5	3, 4	e1a 44747	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐶 ⊆ 𝐷   )
6		idn3 44735	. . . . . . . 8 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
7		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → 𝐴 ⊆ 𝐵)
8	3, 7	e1a 44747	. . . . . . . . . . 11 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐴 ⊆ 𝐵   )
9		idn2 44733	. . . . . . . . . . 11 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   )
10		ssralv 3999	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑))
11	8, 9, 10	e12 44843	. . . . . . . . . 10 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑   )
12		df-ral 3049	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
13	12	biimpi 216	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
14	11, 13	e2 44751	. . . . . . . . 9 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)   )
15		sp 2188	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
16	14, 15	e2 44751	. . . . . . . 8 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)   )
17		pm2.27 42	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑) → ∀𝑦 ∈ 𝐷 𝜑))
18	6, 16, 17	e32 44877	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐷 𝜑   )
19		ssralv 3999	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐷 → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑))
20	5, 18, 19	e13 44867	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐶 𝜑   )
21	20	in3 44729	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑)   )
22	1, 2, 21	gen21nv 44740	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑)   )
23		df-ral 3049	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑))
24	23	biimpri 228	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)
25	22, 24	e2 44751	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑   )
26	25	in2 44725	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)   )
27	26	in1 44691	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-ex 1781  df-nf 1785  df-ral 3049  df-ss 3915  df-vd1 44690  df-vd2 44698  df-vd3 44710

This theorem is referenced by: (None)