ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3orbi123d GIF version

Theorem 3orbi123d 1322
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
Hypotheses
Ref Expression
bi3d.1 (𝜑 → (𝜓𝜒))
bi3d.2 (𝜑 → (𝜃𝜏))
bi3d.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
3orbi123d (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Proof of Theorem 3orbi123d
StepHypRef Expression
1 bi3d.1 . . . 4 (𝜑 → (𝜓𝜒))
2 bi3d.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2orbi12d 794 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 bi3d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4orbi12d 794 . 2 (𝜑 → (((𝜓𝜃) ∨ 𝜂) ↔ ((𝜒𝜏) ∨ 𝜁)))
6 df-3or 981 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
7 df-3or 981 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∨ 𝜁))
85, 6, 73bitr4g 223 1 (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709  w3o 979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710
This theorem depends on definitions:  df-bi 117  df-3or 981
This theorem is referenced by:  ordtriexmid  4535  ontriexmidim  4536  wetriext  4591  nntri3or  6512  tridc  6917  exmidontriimlem3  7240  exmidontriimlem4  7241  exmidontriim  7242  onntri35  7254  ltsopi  7337  pitri3or  7339  nqtri3or  7413  elz  9273  ztri3or  9314  qtri3or  10261  trilpo  15189  trirec0  15190  reap0  15204
  Copyright terms: Public domain W3C validator