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

Theorem anddi 733
 Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
anddi (((φ ψ) (χ θ)) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))

Proof of Theorem anddi
StepHypRef Expression
1 andir 731 . 2 (((φ ψ) (χ θ)) ↔ ((φ (χ θ)) (ψ (χ θ))))
2 andi 730 . . 3 ((φ (χ θ)) ↔ ((φ χ) (φ θ)))
3 andi 730 . . 3 ((ψ (χ θ)) ↔ ((ψ χ) (ψ θ)))
42, 3orbi12i 680 . 2 (((φ (χ θ)) (ψ (χ θ))) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))
51, 4bitri 173 1 (((φ ψ) (χ θ)) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))
 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:  funun  4887  acexmidlemcase  5450  nnm00  6038
 Copyright terms: Public domain W3C validator