Theorem rspc2gv 3546

Description: Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)

Hypothesis

Ref	Expression
rspc2gv.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspc2gv	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rspc2gv

Step	Hyp	Ref	Expression
1		df-ral 3066	. 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑))
2		df-ral 3066	. . . . 5 ⊢ (∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝑊 → 𝜑))
3	2	imbi2i 339	. . . 4 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
4	3	albii 1827	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
5		19.21v 1947	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
6	5	bicomi 227	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
7	6	albii 1827	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
8		impexp 454	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
9		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
10		eleq1 2825	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
11	9, 10	bi2anan9 639	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)))
12		rspc2gv.1	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
13	11, 12	imbi12d 348	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
14	8, 13	bitr3id 288	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
15	14	spc2gv 3515	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
16	15	pm2.43a 54	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
17	7, 16	syl5bi 245	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
18	4, 17	syl5bi 245	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) → 𝜓))
19	1, 18	syl5bi 245	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543   ∈ wcel 2110  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066

This theorem is referenced by:  eulplig  28566  prmidlc  31338  irrdiff  35231  ichreuopeq  44598  isomuspgrlem2b  44954  iscnrm3lem5  45904  iscnrm3r  45915  catprslem  45964  thincmoALT  45984  functhinclem2  45996  fullthinc2  46001  mndtcbas2  46041