MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3orbi123i Structured version   Visualization version   GIF version

Theorem 3orbi123i 1154
Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
Hypotheses
Ref Expression
bi3.1 (𝜑𝜓)
bi3.2 (𝜒𝜃)
bi3.3 (𝜏𝜂)
Assertion
Ref Expression
3orbi123i ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))

Proof of Theorem 3orbi123i
StepHypRef Expression
1 bi3.1 . . . 4 (𝜑𝜓)
2 bi3.2 . . . 4 (𝜒𝜃)
31, 2orbi12i 911 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
4 bi3.3 . . 3 (𝜏𝜂)
53, 4orbi12i 911 . 2 (((𝜑𝜒) ∨ 𝜏) ↔ ((𝜓𝜃) ∨ 𝜂))
6 df-3or 1086 . 2 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 1086 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
85, 6, 73bitr4i 302 1 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 843  w3o 1084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206  df-or 844  df-3or 1086
This theorem is referenced by:  ne3anior  3034  otthne  5485  brtp  5522  wecmpep  5667  cnvso  6286  sorpss  7720  epweon  7764  epweonALT  7765  soxp  8117  dford2  9617  elfz0lmr  13751  sltsolem1  27414  axlowdimlem6  28472  elxrge02  32365  dfon2  35068  frege129d  42816  dfxlim2  44862
  Copyright terms: Public domain W3C validator