Theorem or4 687

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 682	. . 3 ⊢ ((ψ ∨ (χ ∨ θ)) ↔ (χ ∨ (ψ ∨ θ)))
2	1	orbi2i 678	. 2 ⊢ ((φ ∨ (ψ ∨ (χ ∨ θ))) ↔ (φ ∨ (χ ∨ (ψ ∨ θ))))
3		orass 683	. 2 ⊢ (((φ ∨ ψ) ∨ (χ ∨ θ)) ↔ (φ ∨ (ψ ∨ (χ ∨ θ))))
4		orass 683	. 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 628

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 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  or42  688  orordi  689  orordir  690  3or6  1217  swoer  6070  apcotr  7391