Theorem or4 955

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 949	. . 3 ⊢ ((𝜓 ∨ (𝜒 ∨ 𝜃)) ↔ (𝜒 ∨ (𝜓 ∨ 𝜃)))
2	1	orbi2i 941	. 2 ⊢ ((𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
3		orass 950	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))))
4		orass 950	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
5	2, 3, 4	3bitr4i 295	1 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))

Syntax hints:   ↔ wb 198   ∨ wo 878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 199  df-or 879

This theorem is referenced by:  or42  956  orordi  957  orordir  958  3or6  1575  swoer  8044  xmullem2  12390  swrdnnn0nd  13728  clsk1indlem3  39176