Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  spc2d Structured version   Visualization version   GIF version

Theorem spc2d 3582
 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 1828 . . 3 (¬ ∀𝑥𝑦𝜓 ↔ ∃𝑥𝑦 ¬ 𝜓)
21con1bii 359 . 2 (¬ ∃𝑥𝑦 ¬ 𝜓 ↔ ∀𝑥𝑦𝜓)
3 spc2ed.x . . . . 5 𝑥𝜒
43nfn 1857 . . . 4 𝑥 ¬ 𝜒
5 spc2ed.y . . . . 5 𝑦𝜒
65nfn 1857 . . . 4 𝑦 ¬ 𝜒
7 spc2ed.1 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
87notbid 320 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (¬ 𝜓 ↔ ¬ 𝜒))
94, 6, 8spc2ed 3581 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (¬ 𝜒 → ∃𝑥𝑦 ¬ 𝜓))
109con1d 147 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (¬ ∃𝑥𝑦 ¬ 𝜓𝜒))
112, 10syl5bir 245 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-cleq 2813  df-clel 2891 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator