MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.32 Structured version   Visualization version   GIF version

Theorem pm5.32 563
Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
pm5.32 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem pm5.32
StepHypRef Expression
1 notbi 308 . . . 4 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
21imbi2i 325 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)))
3 pm5.74 259 . . 3 ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)))
4 notbi 308 . . 3 (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
52, 3, 43bitri 286 . 2 ((𝜑 → (𝜓𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
6 df-an 383 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
7 df-an 383 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑 → ¬ 𝜒))
86, 7bibi12i 328 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
95, 8bitr4i 267 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383
This theorem is referenced by:  pm5.32i  564  pm5.32d  566  xordi  997  rabbi  3269  rabxfrd  5017  asymref  5653  mpt22eqb  6916  cfilucfil4  23337  wl-ax11-lem8  33703  relexp0eq  38519  2sb5nd  39301  2sb5ndVD  39668  2sb5ndALT  39690
  Copyright terms: Public domain W3C validator