Theorem orass 510

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 376	. 2 ⊢ (((φ ∨ ψ) ∨ χ) ↔ (χ ∨ (φ ∨ ψ)))
2		or12 509	. 2 ⊢ ((χ ∨ (φ ∨ ψ)) ↔ (φ ∨ (χ ∨ ψ)))
3		orcom 376	. . 3 ⊢ ((χ ∨ ψ) ↔ (ψ ∨ χ))
4	3	orbi2i 505	. 2 ⊢ ((φ ∨ (χ ∨ ψ)) ↔ (φ ∨ (ψ ∨ χ)))
5	1, 2, 4	3bitri 262	1 ⊢ (((φ ∨ ψ) ∨ χ) ↔ (φ ∨ (ψ ∨ χ)))

This proof depends on syntax axioms:   ↔ wb 176   ∨ wo 357

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

This proof depends on definitions:  df-bi 177  df-or 359

This theorem is used by:  pm2.31  511  pm2.32  512  or32  513  or4  514  3orass  937  axi12  2333  unass  3421