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

Theorem cbvral3v 2639
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 2434 . . 3 (𝑥 = 𝑤 → (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜒))
32cbvralv 2629 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒)
4 cbvral3v.2 . . . 4 (𝑦 = 𝑣 → (𝜒𝜃))
5 cbvral3v.3 . . . 4 (𝑧 = 𝑢 → (𝜃𝜓))
64, 5cbvral2v 2637 . . 3 (∀𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑣𝐵𝑢𝐶 𝜓)
76ralbii 2416 . 2 (∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
83, 7bitri 183 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator