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Theorem spc2d 3600
Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
spc2ed.x 𝑥𝜒
spc2ed.y 𝑦𝜒
spc2ed.1 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
spc2d ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2d
StepHypRef Expression
1 2nalexn 1819 . . 3 (¬ ∀𝑥𝑦𝜓 ↔ ∃𝑥𝑦 ¬ 𝜓)
21con1bii 358 . 2 (¬ ∃𝑥𝑦 ¬ 𝜓 ↔ ∀𝑥𝑦𝜓)
3 spc2ed.x . . . . 5 𝑥𝜒
43nfn 1848 . . . 4 𝑥 ¬ 𝜒
5 spc2ed.y . . . . 5 𝑦𝜒
65nfn 1848 . . . 4 𝑦 ¬ 𝜒
7 spc2ed.1 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
87notbid 319 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (¬ 𝜓 ↔ ¬ 𝜒))
94, 6, 8spc2ed 3599 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (¬ 𝜒 → ∃𝑥𝑦 ¬ 𝜓))
109con1d 147 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (¬ ∃𝑥𝑦 ¬ 𝜓𝜒))
112, 10syl5bir 244 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wnf 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890
This theorem is referenced by: (None)
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