Theorem or12 217

Description: A rearrangement of disjuncts.

Assertion

Ref	Expression
or12	⊢ ((φ ∨ (ψ ∨ χ)) ↔ (ψ ∨ (φ ∨ χ)))

Proof of Theorem or12

Step	Hyp	Ref	Expression
1		bi2.04 141	. . 3 ⊢ ((¬ ψ → (¬ φ → χ)) ↔ (¬ φ → (¬ ψ → χ)))
2		df-or 197	. . . 4 ⊢ ((φ ∨ χ) ↔ (¬ φ → χ))
3	2	imbi2i 160	. . 3 ⊢ ((¬ ψ → (φ ∨ χ)) ↔ (¬ ψ → (¬ φ → χ)))
4		df-or 197	. . . 4 ⊢ ((ψ ∨ χ) ↔ (¬ ψ → χ))
5	4	imbi2i 160	. . 3 ⊢ ((¬ φ → (ψ ∨ χ)) ↔ (¬ φ → (¬ ψ → χ)))
6	1, 3, 5	3bitr4r 159	. 2 ⊢ ((¬ φ → (ψ ∨ χ)) ↔ (¬ ψ → (φ ∨ χ)))
7		df-or 197	. 2 ⊢ ((φ ∨ (ψ ∨ χ)) ↔ (¬ φ → (ψ ∨ χ)))
8		df-or 197	. 2 ⊢ ((ψ ∨ (φ ∨ χ)) ↔ (¬ ψ → (φ ∨ χ)))
9	6, 7, 8	3bitr4 158	1 ⊢ ((φ ∨ (ψ ∨ χ)) ↔ (ψ ∨ (φ ∨ χ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195

This theorem is referenced by:  orass 218  or4 220  ordzsl 2366  posex 4422

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197

metamath.org