Theorem 3orbi123i 1156

Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)

Hypotheses

Ref	Expression
bi3.1	⊢ (𝜑 ↔ 𝜓)
bi3.2	⊢ (𝜒 ↔ 𝜃)
bi3.3	⊢ (𝜏 ↔ 𝜂)

Assertion

Ref	Expression
3orbi123i	⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))

Proof of Theorem 3orbi123i

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2		bi3.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
3	1, 2	orbi12i 913	. . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))
4		bi3.3	. . 3 ⊢ (𝜏 ↔ 𝜂)
5	3, 4	orbi12i 913	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
6		df-3or 1088	. 2 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜏))
7		df-3or 1088	. 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
8	5, 6, 7	3bitr4i 302	1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))

Syntax hints:   ↔ wb 205   ∨ wo 845   ∨ w3o 1086

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 846  df-3or 1088

This theorem is referenced by:  ne3anior  3035  otthne  5448  brtp  5485  wecmpep  5630  cnvso  6245  sorpss  7670  epweon  7714  epweonALT  7715  soxp  8066  dford2  9565  elfz0lmr  13697  sltsolem1  27060  axlowdimlem6  27959  elxrge02  31858  dfon2  34453  frege129d  42157  dfxlim2  44209