ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordi Structured version   GIF version

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
StepHypRef Expression
1 simpl 102 . . . 4 ((ψ χ) → ψ)
21orim2i 677 . . 3 ((φ (ψ χ)) → (φ ψ))
3 simpr 103 . . . 4 ((ψ χ) → χ)
43orim2i 677 . . 3 ((φ (ψ χ)) → (φ χ))
52, 4jca 290 . 2 ((φ (ψ χ)) → ((φ ψ) (φ χ)))
6 orc 632 . . . 4 (φ → (φ (ψ χ)))
76adantl 262 . . 3 (((φ ψ) φ) → (φ (ψ χ)))
86adantr 261 . . . 4 ((φ χ) → (φ (ψ χ)))
9 olc 631 . . . 4 ((ψ χ) → (φ (ψ χ)))
108, 9jaoian 708 . . 3 (((φ ψ) χ) → (φ (ψ χ)))
117, 10jaodan 709 . 2 (((φ ψ) (φ χ)) → (φ (ψ χ)))
125, 11impbii 117 1 ((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator