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Theorem spc2gv 3400
 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 307 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝜑 ↔ ¬ 𝜓))
32spc2egv 3399 . . 3 ((𝐴𝑉𝐵𝑊) → (¬ 𝜓 → ∃𝑥𝑦 ¬ 𝜑))
4 2nalexn 1868 . . 3 (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
53, 4syl6ibr 242 . 2 ((𝐴𝑉𝐵𝑊) → (¬ 𝜓 → ¬ ∀𝑥𝑦𝜑))
65con4d 114 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1594   = wceq 1596  ∃wex 1817   ∈ wcel 2103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-v 3306 This theorem is referenced by:  rspc2gv  3425  trel  4867  elovmpt2  6996  seqf1olem2  12956  seqf1o  12957  fi1uzind  13392  brfi1indALT  13395  pslem  17328  cnmpt12  21593  cnmpt22  21600  mclsppslem  31708  mbfresfi  33688  lpolconN  37195  ismrcd2  37681  ismrc  37683
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