Theorem ssralv2VD 41206

Description: Quantification restricted to a subclass for two quantifiers. ssralv 4036 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 40871 is ssralv2VD 41206 without virtual deductions and was automatically derived from ssralv2VD 41206.

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 1910	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
2		hbra1 3223	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑)
3		idn1 40914	. . . . . . . 8 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   )
4		simpr 487	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → 𝐶 ⊆ 𝐷)
5	3, 4	e1a 40967	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐶 ⊆ 𝐷   )
6		idn3 40955	. . . . . . . 8 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
7		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → 𝐴 ⊆ 𝐵)
8	3, 7	e1a 40967	. . . . . . . . . . 11 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐴 ⊆ 𝐵   )
9		idn2 40953	. . . . . . . . . . 11 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   )
10		ssralv 4036	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑))
11	8, 9, 10	e12 41064	. . . . . . . . . 10 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑   )
12		df-ral 3146	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
13	12	biimpi 218	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
14	11, 13	e2 40971	. . . . . . . . 9 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)   )
15		sp 2181	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))
16	14, 15	e2 40971	. . . . . . . 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 41098	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐷 𝜑   )
19		ssralv 4036	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐷 → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑))
20	5, 18, 19	e13 41088	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ,   𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐶 𝜑   )
21	20	in3 40949	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑)   )
22	1, 2, 21	gen21nv 40960	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑)   )
23		df-ral 3146	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑))
24	23	biimpri 230	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)
25	22, 24	e2 40971	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑   )
26	25	in2 40945	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)   )
27	26	in1 40911	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534   ∈ wcel 2113  ∀wral 3141   ⊆ wss 3939

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3146  df-in 3946  df-ss 3955  df-vd1 40910  df-vd2 40918  df-vd3 40930

This theorem is referenced by: (None)