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

Theorem 3orbi123i 1157
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 914 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
4 bi3.3 . . 3 (𝜏𝜂)
53, 4orbi12i 914 . 2 (((𝜑𝜒) ∨ 𝜏) ↔ ((𝜓𝜃) ∨ 𝜂))
6 df-3or 1089 . 2 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 1089 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
85, 6, 73bitr4i 303 1 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 846  w3o 1087
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 847  df-3or 1089
This theorem is referenced by:  ne3anior  3037  otthne  5487  brtp  5524  wecmpep  5669  cnvso  6288  sorpss  7718  epweon  7762  epweonALT  7763  soxp  8115  dford2  9615  elfz0lmr  13747  sltsolem1  27178  axlowdimlem6  28205  elxrge02  32098  dfon2  34764  frege129d  42514  dfxlim2  44564
  Copyright terms: Public domain W3C validator