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Theorem cbvral3v 2845
 Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
cbvral3v.1 (x = w → (φχ))
cbvral3v.2 (y = v → (χθ))
cbvral3v.3 (z = u → (θψ))
Assertion
Ref Expression
cbvral3v (x A y B z C φw A v B u C ψ)
Distinct variable groups:   φ,w   ψ,z   χ,x   χ,v   y,u,θ   x,A   w,A   x,y,B   y,w,B   v,B   x,z,C,y   z,w,C   z,v,C   u,C
Allowed substitution hints:   φ(x,y,z,v,u)   ψ(x,y,w,v,u)   χ(y,z,w,u)   θ(x,z,w,v)   A(y,z,v,u)   B(z,u)

Proof of Theorem cbvral3v
StepHypRef Expression
1 cbvral3v.1 . . . 4 (x = w → (φχ))
212ralbidv 2656 . . 3 (x = w → (y B z C φy B z C χ))
32cbvralv 2835 . 2 (x A y B z C φw A y B z C χ)
4 cbvral3v.2 . . . 4 (y = v → (χθ))
5 cbvral3v.3 . . . 4 (z = u → (θψ))
64, 5cbvral2v 2843 . . 3 (y B z C χv B u C ψ)
76ralbii 2638 . 2 (w A y B z C χw A v B u C ψ)
83, 7bitri 240 1 (x A y B z C φw A v B u C ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀wral 2614 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-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by: (None)
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