Theorem rspc2gv 3321

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 2917	. 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑))
2		df-ral 2917	. . . . 5 ⊢ (∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝑊 → 𝜑))
3	2	imbi2i 326	. . . 4 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
4	3	albii 1747	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
5		19.21v 1868	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
6	5	bicomi 214	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
7	6	albii 1747	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
8		impexp 462	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
9		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
10		eleq1 2689	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
11	9, 10	bi2anan9 917	. . . . . . . 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 334	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
14	8, 13	syl5bbr 274	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
15	14	spc2gv 3296	. . . . 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 232	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
18	4, 17	syl5bi 232	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) → 𝜓))
19	1, 18	syl5bi 232	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202

This theorem is referenced by:  eulplig  27337