Theorem or4 688

Description: Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
or4	⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))

Proof of Theorem or4

Step	Hyp	Ref	Expression
1		or12 683	. . 3 ⊢ ((𝜓 ∨ (𝜒 ∨ 𝜃)) ↔ (𝜒 ∨ (𝜓 ∨ 𝜃)))
2	1	orbi2i 679	. 2 ⊢ ((𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
3		orass 684	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))))
4		orass 684	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
5	2, 3, 4	3bitr4i 201	1 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))

Syntax hints:   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  or42  689  orordi  690  orordir  691  3or6  1218  swoer  6134  apcotr  7598