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

Theorem jcab 545
 Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
Assertion
Ref Expression
jcab ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem jcab
StepHypRef Expression
1 simpl 106 . . . 4 ((𝜓𝜒) → 𝜓)
21imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
3 simpr 107 . . . 4 ((𝜓𝜒) → 𝜒)
43imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 294 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 pm3.43 544 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
75, 6impbii 121 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  pm4.76  546  pm5.44  845  2eu4  2009  ssconb  3103  ssin  3186  raaan  3354  tfri3  5983
 Copyright terms: Public domain W3C validator