Theorem orass 935

Description: Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
orass	⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))

Proof of Theorem orass

Step	Hyp	Ref	Expression
1		orcom 884	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∨ 𝜓)))
2		or12 934	. 2 ⊢ ((𝜒 ∨ (𝜑 ∨ 𝜓)) ↔ (𝜑 ∨ (𝜒 ∨ 𝜓)))
3		orcom 884	. . 3 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
4	3	orbi2i 926	. 2 ⊢ ((𝜑 ∨ (𝜒 ∨ 𝜓)) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
5	1, 2, 4	3bitri 300	1 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by:  pm2.31  936  pm2.32  937  or32  939  or4  940  3orass  1106  axi12  2733  axbnd  2734  unass  4173  tppreqb  4770  ltxr  13284  lcmass  16833  plydivex  26673  clwwlkneq0  30677  disjxpin  33280  wl-ifpimpr  38657  impor  39281  ifpim123g  44762