ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvral3v GIF version

Theorem cbvral3v 2560
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 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 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 2365 . . 3 (𝑥 = 𝑤 → (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜒))
32cbvralv 2550 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒)
4 cbvral3v.2 . . . 4 (𝑦 = 𝑣 → (𝜒𝜃))
5 cbvral3v.3 . . . 4 (𝑧 = 𝑢 → (𝜃𝜓))
64, 5cbvral2v 2558 . . 3 (∀𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑣𝐵𝑢𝐶 𝜓)
76ralbii 2347 . 2 (∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
83, 7bitri 177 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator