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Theorem notbi 319
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 318 . 2 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3 id 22 . . 3 ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43con4bid 317 . 2 ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑𝜓))
52, 4impbii 208 1 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205
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 206
This theorem is referenced by:  notbii  320  con4bii  321  con2bi  354  nbn2  371  pm5.32  575  norass  1539  hadnot  1604  had0  1606  cbvexdw  2336  cbvexd  2408  rexbiOLD  3106  rexprg  4701  isocnv3  7329  suppimacnv  8159  sumodd  16331  f1omvdco3  19317  ist0cld  32813  onsuct0  35326  bj-cbvexdv  35678  wl-3xornot  36362  ifpbi1  42228  ifpbi13  42240  abciffcbatnabciffncba  45639  abciffcbatnabciffncbai  45640  ichn  46124
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