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Theorem spc3gv 2944
 Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1 ((x = A y = B z = C) → (φψ))
Assertion
Ref Expression
spc3gv ((A V B W C X) → (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 = A y = B z = C) → (φψ))
21notbid 285 . . . 4 ((x = A y = B z = C) → (¬ φ ↔ ¬ ψ))
32spc3egv 2943 . . 3 ((A V B W C X) → (¬ ψxyz ¬ φ))
4 exnal 1574 . . . . . . 7 (z ¬ φ ↔ ¬ zφ)
54exbii 1582 . . . . . 6 (yz ¬ φy ¬ zφ)
6 exnal 1574 . . . . . 6 (y ¬ zφ ↔ ¬ yzφ)
75, 6bitri 240 . . . . 5 (yz ¬ φ ↔ ¬ yzφ)
87exbii 1582 . . . 4 (xyz ¬ φx ¬ yzφ)
9 exnal 1574 . . . 4 (x ¬ yzφ ↔ ¬ xyzφ)
108, 9bitr2i 241 . . 3 xyzφxyz ¬ φ)
113, 10syl6ibr 218 . 2 ((A V B W C X) → (¬ ψ → ¬ xyzφ))
1211con4d 97 1 ((A V B W C X) → (xyzφψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  fununiq  5517
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