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

Theorem notbi 309
Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
Assertion
Ref Expression
notbi ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem notbi
StepHypRef Expression
1 id 22 . . 3 ((𝜑𝜓) → (𝜑𝜓))
21notbid 308 . 2 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3 id 22 . . 3 ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43con4bid 307 . 2 ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑𝜓))
52, 4impbii 199 1 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196
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
This theorem is referenced by:  notbii  310  con4bii  311  con2bi  343  nbn2  360  pm5.32  667  hadnot  1538  had0  1540  cbvexd  2282  symdifass  3836  isocnv3  6537  suppimacnv  7252  sumodd  15030  f1omvdco3  17785  onsuct0  32055  bj-cbvexdv  32351  ifpbi1  37270  ifpbi13  37282  abciffcbatnabciffncba  40368  abciffcbatnabciffncbai  40369
  Copyright terms: Public domain W3C validator