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Theorem cbvral3v 2537
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 2342 . . 3 (x = w → (y B z 𝐶 φy B z 𝐶 χ))
32cbvralv 2527 . 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 2535 . . 3 (y B z 𝐶 χv B u 𝐶 ψ)
76ralbii 2324 . 2 (w A y B z 𝐶 χw A v B u 𝐶 ψ)
83, 7bitri 173 1 (x A y B z 𝐶 φw A v B u 𝐶 ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305
This theorem is referenced by: (None)
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