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Theorem rspc2gv 3452
Description: Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
Hypothesis
Ref Expression
rspc2gv.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
rspc2gv ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
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
StepHypRef Expression
1 df-ral 3047 . 2 (∀𝑥𝑉𝑦𝑊 𝜑 ↔ ∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑))
2 df-ral 3047 . . . . 5 (∀𝑦𝑊 𝜑 ↔ ∀𝑦(𝑦𝑊𝜑))
32imbi2i 325 . . . 4 ((𝑥𝑉 → ∀𝑦𝑊 𝜑) ↔ (𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
43albii 1888 . . 3 (∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑) ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
5 19.21v 2009 . . . . . 6 (∀𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) ↔ (𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
65bicomi 214 . . . . 5 ((𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) ↔ ∀𝑦(𝑥𝑉 → (𝑦𝑊𝜑)))
76albii 1888 . . . 4 (∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) ↔ ∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)))
8 impexp 461 . . . . . . 7 (((𝑥𝑉𝑦𝑊) → 𝜑) ↔ (𝑥𝑉 → (𝑦𝑊𝜑)))
9 eleq1 2819 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝑉𝐴𝑉))
10 eleq1 2819 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
119, 10bi2anan9 953 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑉𝑦𝑊) ↔ (𝐴𝑉𝐵𝑊)))
12 rspc2gv.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1311, 12imbi12d 333 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑉𝑦𝑊) → 𝜑) ↔ ((𝐴𝑉𝐵𝑊) → 𝜓)))
148, 13syl5bbr 274 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑉 → (𝑦𝑊𝜑)) ↔ ((𝐴𝑉𝐵𝑊) → 𝜓)))
1514spc2gv 3428 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) → ((𝐴𝑉𝐵𝑊) → 𝜓)))
1615pm2.43a 54 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) → 𝜓))
177, 16syl5bi 232 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) → 𝜓))
184, 17syl5bi 232 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑) → 𝜓))
191, 18syl5bi 232 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1622   = wceq 1624  wcel 2131  wral 3042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-ral 3047  df-v 3334
This theorem is referenced by:  eulplig  27640
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