Theorem rspc2gv 3586

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 3078	. 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑))
2		df-ral 3078	. . . . 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 1852	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
5		19.21v 1972	. . . . . 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 1852	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
8		impexp 456	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
9		eleq1 2849	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
10		eleq1 2849	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
11	9, 10	bi2anan9 650	. . . . . . . 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 347	. . . . . . 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 3555	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
16	15	pm2.43a 55	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
17	7, 16	biimtrid 245	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
18	4, 17	biimtrid 245	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) → 𝜓))
19	1, 18	biimtrid 245	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078

This theorem is used by:  onelfvnef1  8449  prmidlc  21629  eulplig  31087  irrdiff  38247  ichreuopeq  48554  isuspgrimlem  48992  isubgr3stgrlem4  49066  iscnrm3lem5  50044  iscnrm3r  50055  catprslem  50117  oppcendc  50125  thincmoALT  50536  functhinclem2  50552  fullthinc2  50558  mndtcobeq  50690