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Theorem notbi 308
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 307 . 2 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3 id 22 . . 3 ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43con4bid 306 . 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  309  con4bii  310  con2bi  342  nbn2  359  pm5.32  669  hadnot  1581  had0  1583  cbvexd  2314  symdifass  3886  isocnv3  6622  suppimacnv  7351  sumodd  15158  f1omvdco3  17915  onsuct0  32565  bj-cbvexdv  32861  ifpbi1  38139  ifpbi13  38151  abciffcbatnabciffncba  41417  abciffcbatnabciffncbai  41418
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