Theorem rspc2gv 2846

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 2453	. 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑))
2		df-ral 2453	. . . . 5 ⊢ (∀𝑦 ∈ 𝑊 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝑊 → 𝜑))
3	2	imbi2i 225	. . . 4 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
4	3	albii 1463	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
5		19.21v 1866	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)))
6	5	bicomi 131	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
7	6	albii 1463	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
8		impexp 261	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)))
9		eleq1 2233	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
10		eleq1 2233	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
11	9, 10	bi2anan9 601	. . . . . . . 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 233	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → 𝜑) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
14	8, 13	bitr3id 193	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
15	14	spc2gv 2821	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜓)))
16	15	pm2.43a 51	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦(𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
17	7, 16	syl5bi 151	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑊 → 𝜑)) → 𝜓))
18	4, 17	syl5bi 151	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝑊 𝜑) → 𝜓))
19	1, 18	syl5bi 151	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348   ∈ wcel 2141  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-v 2732

This theorem is referenced by:  difinfsnlem  7076  difinfsn  7077  seqvalcd  10415  seqovcd  10419  qtopbasss  13315  apdiff  14080