Theorem or4 550

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 545	. . 3 ⊢ ((𝜓 ∨ (𝜒 ∨ 𝜃)) ↔ (𝜒 ∨ (𝜓 ∨ 𝜃)))
2	1	orbi2i 541	. 2 ⊢ ((𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
3		orass 546	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜓 ∨ (𝜒 ∨ 𝜃))))
4		orass 546	. 2 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) ↔ (𝜑 ∨ (𝜒 ∨ (𝜓 ∨ 𝜃))))
5	2, 3, 4	3bitr4i 292	1 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))

Syntax hints:   ↔ wb 196   ∨ wo 383

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 197  df-or 385

This theorem is referenced by:  or42  551  orordi  552  orordir  553  3or6  1410  swoer  7772  xmullem2  12095  clsk1indlem3  38341