Theorem ordi 452

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119.

Assertion

Ref	Expression
ordi	⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ)))

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . 4 ⊢ ((ψ ∧ χ) → ψ)
2	1	orim2i 273	. . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ ψ))
3		pm3.27 260	. . . 4 ⊢ ((ψ ∧ χ) → χ)
4	3	orim2i 273	. . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ χ))
5	2, 4	jca 236	. 2 ⊢ ((φ ∨ (ψ ∧ χ)) → ((φ ∨ ψ) ∧ (φ ∨ χ)))
6		df-or 197	. . . 4 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
7		pm3.43i 235	. . . . 5 ⊢ ((¬ φ → ψ) → ((¬ φ → χ) → (¬ φ → (ψ ∧ χ))))
8		df-or 197	. . . . 5 ⊢ ((φ ∨ χ) ↔ (¬ φ → χ))
9		df-or 197	. . . . 5 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ (¬ φ → (ψ ∧ χ)))
10	7, 8, 9	3imtr4g 426	. . . 4 ⊢ ((¬ φ → ψ) → ((φ ∨ χ) → (φ ∨ (ψ ∧ χ))))
11	6, 10	sylbi 174	. . 3 ⊢ ((φ ∨ ψ) → ((φ ∨ χ) → (φ ∨ (ψ ∧ χ))))
12	11	imp 277	. 2 ⊢ (((φ ∨ ψ) ∧ (φ ∨ χ)) → (φ ∨ (ψ ∧ χ)))
13	5, 12	impbi 139	1 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  ordir 453  jcab 454  andi 456  orddi 458  orbidi 510  undi 1677  undif4 1744

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  df-an 198

metamath.org