ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r3al GIF version

Theorem r3al 2420
Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
Assertion
Ref Expression
r3al (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑧,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem r3al
StepHypRef Expression
1 df-ral 2364 . 2 (∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
2 r2al 2397 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
32ralbii 2384 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
4 3anass 928 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)))
54imbi1i 236 . . . . . . . 8 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑))
6 impexp 259 . . . . . . . 8 (((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
75, 6bitri 182 . . . . . . 7 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
87albii 1404 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
9 19.21v 1801 . . . . . 6 (∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
108, 9bitri 182 . . . . 5 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1110albii 1404 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
12 19.21v 1801 . . . 4 (∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1311, 12bitri 182 . . 3 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1413albii 1404 . 2 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
151, 3, 143bitr4i 210 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924  wal 1287  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364
This theorem is referenced by:  pocl  4121  soss  4132
  Copyright terms: Public domain W3C validator