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Theorem spc3egv 2943
 Description: Existential 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
spc3egv ((A V B W C X) → (ψxyzφ))
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 spc3egv
StepHypRef Expression
1 elisset 2869 . . . 4 (A Vx x = A)
2 elisset 2869 . . . 4 (B Wy y = B)
3 elisset 2869 . . . 4 (C Xz z = C)
41, 2, 33anim123i 1137 . . 3 ((A V B W C X) → (x x = A y y = B z z = C))
5 eeeanv 1914 . . 3 (xyz(x = A y = B z = C) ↔ (x x = A y y = B z z = C))
64, 5sylibr 203 . 2 ((A V B W C X) → xyz(x = A y = B z = C))
7 spc3egv.1 . . . . 5 ((x = A y = B z = C) → (φψ))
87biimprcd 216 . . . 4 (ψ → ((x = A y = B z = C) → φ))
98eximdv 1622 . . 3 (ψ → (z(x = A y = B z = C) → zφ))
1092eximdv 1624 . 2 (ψ → (xyz(x = A y = B z = C) → xyzφ))
116, 10syl5com 26 1 ((A V B W C X) → (ψxyzφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∃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:  spc3gv  2944
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