Theorem or4 514

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 509	. . 3 ⊢ ((ψ ∨ (χ ∨ θ)) ↔ (χ ∨ (ψ ∨ θ)))
2	1	orbi2i 505	. 2 ⊢ ((φ ∨ (ψ ∨ (χ ∨ θ))) ↔ (φ ∨ (χ ∨ (ψ ∨ θ))))
3		orass 510	. 2 ⊢ (((φ ∨ ψ) ∨ (χ ∨ θ)) ↔ (φ ∨ (ψ ∨ (χ ∨ θ))))
4		orass 510	. 2 ⊢ (((φ ∨ χ) ∨ (ψ ∨ θ)) ↔ (φ ∨ (χ ∨ (ψ ∨ θ))))
5	2, 3, 4	3bitr4i 268	1 ⊢ (((φ ∨ ψ) ∨ (χ ∨ θ)) ↔ ((φ ∨ χ) ∨ (ψ ∨ θ)))

This proof depends on syntax axioms:   ↔ wb 176   ∨ wo 357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359

This theorem is used by:  or42  515  orordi  516  orordir  517  3or6  1263