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

Theorem pm5.32 441
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 438 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
3 ibar 295 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
4 ibar 295 . . . 4 (𝜑 → (𝜒 ↔ (𝜑𝜒)))
53, 4bibi12d 233 . . 3 (𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
65biimprcd 158 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
72, 6impbii 124 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm5.32i  442  xordidc  1331  cbvex2  1839  rabbi  2532  rabxfrd  4221  asymref  4734  rexrnmpt  5336  mpt22eqb  5635
  Copyright terms: Public domain W3C validator