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Theorem spc2gv 3520
 Description: Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
Hypothesis
Ref Expression
spc2egv.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
spc2gv ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2gv
StepHypRef Expression
1 spc2egv.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
21notbid 322 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝜑 ↔ ¬ 𝜓))
32spc2egv 3519 . . 3 ((𝐴𝑉𝐵𝑊) → (¬ 𝜓 → ∃𝑥𝑦 ¬ 𝜑))
4 2nalexn 1830 . . 3 (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
53, 4syl6ibr 255 . 2 ((𝐴𝑉𝐵𝑊) → (¬ 𝜓 → ¬ ∀𝑥𝑦𝜑))
65con4d 115 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by:  rspc2gv  3551  trel  5143  elovmpo  7384  seqf1olem2  13450  seqf1o  13451  fi1uzind  13897  brfi1indALT  13900  pslem  17872  cnmpt12  22357  cnmpt22  22364  mclsppslem  33051  mbfresfi  35373  lpolconN  39053  ismrcd2  40003  ismrc  40005
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