Theorem or42 767

Description: Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)

Assertion

Ref	Expression
or42	⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓)))

Proof of Theorem or42

Step	Hyp	Ref	Expression
1		or4 766	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))
2		orcom 723	. . 3 ⊢ ((𝜓 ∨ 𝜃) ↔ (𝜃 ∨ 𝜓))
3	2	orbi2i 757	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓)))
4	1, 3	bitri 183	1 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓)))

Syntax hints:   ↔ wb 104   ∨ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  reapcotr  8517