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Theorem rspc2gv 3580
 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 3111 . 2 (∀𝑥𝑉𝑦𝑊 𝜑 ↔ ∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑))
2 df-ral 3111 . . . . 5 (∀𝑦𝑊 𝜑 ↔ ∀𝑦(𝑦𝑊𝜑))
32imbi2i 339 . . . 4 ((𝑥𝑉 → ∀𝑦𝑊 𝜑) ↔ (𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
43albii 1821 . . 3 (∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑) ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
5 19.21v 1940 . . . . . 6 (∀𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) ↔ (𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)))
65bicomi 227 . . . . 5 ((𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) ↔ ∀𝑦(𝑥𝑉 → (𝑦𝑊𝜑)))
76albii 1821 . . . 4 (∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) ↔ ∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)))
8 impexp 454 . . . . . . 7 (((𝑥𝑉𝑦𝑊) → 𝜑) ↔ (𝑥𝑉 → (𝑦𝑊𝜑)))
9 eleq1 2877 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝑉𝐴𝑉))
10 eleq1 2877 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
119, 10bi2anan9 638 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑉𝑦𝑊) ↔ (𝐴𝑉𝐵𝑊)))
12 rspc2gv.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1311, 12imbi12d 348 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑉𝑦𝑊) → 𝜑) ↔ ((𝐴𝑉𝐵𝑊) → 𝜓)))
148, 13bitr3id 288 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑉 → (𝑦𝑊𝜑)) ↔ ((𝐴𝑉𝐵𝑊) → 𝜓)))
1514spc2gv 3549 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) → ((𝐴𝑉𝐵𝑊) → 𝜓)))
1615pm2.43a 54 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦(𝑥𝑉 → (𝑦𝑊𝜑)) → 𝜓))
177, 16syl5bi 245 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑊𝜑)) → 𝜓))
184, 17syl5bi 245 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥𝑉 → ∀𝑦𝑊 𝜑) → 𝜓))
191, 18syl5bi 245 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111 This theorem is referenced by:  eulplig  28278  prmidlc  31042  irrdiff  34759  ichreuopeq  44033  isomuspgrlem2b  44390
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