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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 198 1 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195 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 196 This theorem is referenced by:  notbii  309  con4bii  310  con2bi  342  nbn2  359  pm5.32  666  hadnot  1532  had0  1534  cbvexd  2266  symdifass  3815  isocnv3  6482  suppimacnv  7193  sumodd  14949  f1omvdco3  17692  onsuct0  31610  bj-cbvexdv  31923  ifpbi1  36841  ifpbi13  36853  abciffcbatnabciffncba  39745  abciffcbatnabciffncbai  39746
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