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 914	. . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))
4		bi3.3	. . 3 ⊢ (𝜏 ↔ 𝜂)
5	3, 4	orbi12i 914	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
6		df-3or 1087	. 2 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜏))
7		df-3or 1087	. 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
8	5, 6, 7	3bitr4i 303	1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∨ w3o 1085

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 207  df-or 848  df-3or 1087

This theorem is referenced by:  ne3anior  3025  otthne  5471  brtp  5508  wecmpep  5657  cnvso  6288  sorpss  7730  epweon  7777  epweonALT  7778  soxp  8136  dford2  9642  elfz0lmr  13803  hash3tpde  14514  sltsolem1  27656  axlowdimlem6  28892  elxrge02  32854  constrcbvlem  33735  dfon2  35752  frege129d  43738  dfxlim2  45820  usgrexmpl2trifr  47954