Theorem orass 768

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 729	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∨ 𝜓)))
2		or12 767	. 2 ⊢ ((𝜒 ∨ (𝜑 ∨ 𝜓)) ↔ (𝜑 ∨ (𝜒 ∨ 𝜓)))
3		orcom 729	. . 3 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
4	3	orbi2i 763	. 2 ⊢ ((𝜑 ∨ (𝜒 ∨ 𝜓)) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
5	1, 2, 4	3bitri 206	1 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))

Syntax hints:   ↔ wb 105   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm2.31  769  pm2.32  770  or32  771  or4  772  3orass  983  dveeq2  1829  dveeq2or  1830  sbequilem  1852  dvelimALT  2029  dvelimfv  2030  dvelimor  2037  unass  3320  ltxr  9850  lcmass  12253