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Theorem spc3gv 3296
Description: Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
Assertion
Ref Expression
spc3gv ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem spc3gv
StepHypRef Expression
1 spc3egv.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
21notbid 308 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (¬ 𝜑 ↔ ¬ 𝜓))
32spc3egv 3295 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (¬ 𝜓 → ∃𝑥𝑦𝑧 ¬ 𝜑))
4 exnal 1753 . . . . . . 7 (∃𝑧 ¬ 𝜑 ↔ ¬ ∀𝑧𝜑)
54exbii 1773 . . . . . 6 (∃𝑦𝑧 ¬ 𝜑 ↔ ∃𝑦 ¬ ∀𝑧𝜑)
6 exnal 1753 . . . . . 6 (∃𝑦 ¬ ∀𝑧𝜑 ↔ ¬ ∀𝑦𝑧𝜑)
75, 6bitri 264 . . . . 5 (∃𝑦𝑧 ¬ 𝜑 ↔ ¬ ∀𝑦𝑧𝜑)
87exbii 1773 . . . 4 (∃𝑥𝑦𝑧 ¬ 𝜑 ↔ ∃𝑥 ¬ ∀𝑦𝑧𝜑)
9 exnal 1753 . . . 4 (∃𝑥 ¬ ∀𝑦𝑧𝜑 ↔ ¬ ∀𝑥𝑦𝑧𝜑)
108, 9bitr2i 265 . . 3 (¬ ∀𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧 ¬ 𝜑)
113, 10syl6ibr 242 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (¬ 𝜓 → ¬ ∀𝑥𝑦𝑧𝜑))
1211con4d 114 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  w3a 1037  wal 1480   = wceq 1482  wex 1703  wcel 1989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200
This theorem is referenced by:  funopg  5920  pslem  17200  dirtr  17230  mclsax  31451  fununiq  31653
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