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Theorem spc2gv 2942
 Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
Hypothesis
Ref Expression
spc2egv.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
spc2gv ((A V B W) → (xyφψ))
Distinct variable groups:   x,y,A   x,B,y   ψ,x,y
Allowed substitution hints:   φ(x,y)   V(x,y)   W(x,y)

Proof of Theorem spc2gv
StepHypRef Expression
1 spc2egv.1 . . . . 5 ((x = A y = B) → (φψ))
21notbid 285 . . . 4 ((x = A y = B) → (¬ φ ↔ ¬ ψ))
32spc2egv 2941 . . 3 ((A V B W) → (¬ ψxy ¬ φ))
4 2nalexn 1573 . . 3 xyφxy ¬ φ)
53, 4syl6ibr 218 . 2 ((A V B W) → (¬ ψ → ¬ xyφ))
65con4d 97 1 ((A V B W) → (xyφψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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-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:  nnsucelr  4428  ssfin  4470  ncfinlower  4483
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