Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.32 GIF version

Theorem pm5.32 426
 Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
Assertion
Ref Expression
pm5.32 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem pm5.32
StepHypRef Expression
1 id 19 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓𝜒)))
21pm5.32d 423 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
3 ibar 285 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
4 ibar 285 . . . 4 (𝜑 → (𝜒 ↔ (𝜑𝜒)))
53, 4bibi12d 224 . . 3 (𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
65biimprcd 149 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
72, 6impbii 117 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  pm5.32i  427  xordidc  1290  cbvex2  1797  rabbi  2487  rabxfrd  4201  asymref  4710  rexrnmpt  5310  mpt22eqb  5610
 Copyright terms: Public domain W3C validator