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Theorem notbi 304
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 303 . 2 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3 id 22 . . 3 ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43con4bid 302 . 2 ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑𝜓))
52, 4impbii 194 1 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 191
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 192
This theorem is referenced by:  notbii  305  con4bii  306  con2bi  337  nbn2  354  pm5.32  657  hadnot  1529  had0  1531  cbvexd  2166  symdifass  3699  isocnv3  6296  suppimacnv  7004  f1omvdco3  17251  onsuct0  31250  bj-cbvexdv  31524  ifpbi1  36361  ifpbi13  36373  abciffcbatnabciffncba  39037  abciffcbatnabciffncbai  39038
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