Theorem ordi 728

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		simpl 102	. . . 4 ⊢ ((ψ ∧ χ) → ψ)
2	1	orim2i 677	. . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ ψ))
3		simpr 103	. . . 4 ⊢ ((ψ ∧ χ) → χ)
4	3	orim2i 677	. . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ χ))
5	2, 4	jca 290	. 2 ⊢ ((φ ∨ (ψ ∧ χ)) → ((φ ∨ ψ) ∧ (φ ∨ χ)))
6		orc 632	. . . 4 ⊢ (φ → (φ ∨ (ψ ∧ χ)))
7	6	adantl 262	. . 3 ⊢ (((φ ∨ ψ) ∧ φ) → (φ ∨ (ψ ∧ χ)))
8	6	adantr 261	. . . 4 ⊢ ((φ ∧ χ) → (φ ∨ (ψ ∧ χ)))
9		olc 631	. . . 4 ⊢ ((ψ ∧ χ) → (φ ∨ (ψ ∧ χ)))
10	8, 9	jaoian 708	. . 3 ⊢ (((φ ∨ ψ) ∧ χ) → (φ ∨ (ψ ∧ χ)))
11	7, 10	jaodan 709	. 2 ⊢ (((φ ∨ ψ) ∧ (φ ∨ χ)) → (φ ∨ (ψ ∧ χ)))
12	5, 11	impbii 117	1 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ)))

Syntax hints:   ∧ wa 97   ↔ 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:  ordir  729  orddi  732  pm5.63dc  852  pm4.43  855  orbididc  859  undi  3179  undif4  3278  elnn1uz2  8280