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	⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem notbi

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	notbid 318	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3		id 22	. . 3 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	con4bid 317	. 2 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑 ↔ 𝜓))
5	2, 4	impbii 209	1 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206

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 207

This theorem is referenced by:  notbii  320  con4bii  321  con2bi  353  nbn2  370  pm5.32  573  norass  1537  hadnot  1602  had0  1604  cbvexdw  2337  cbvexd  2406  rexprg  4657  isocnv3  7289  suppimacnv  8130  sumodd  16334  f1omvdco3  19355  ist0cld  33796  onsuct0  36402  bj-cbvexdv  36761  wl-3xornot  37442  ifpbi1  43439  ifpbi13  43451  abciffcbatnabciffncba  46903  abciffcbatnabciffncbai  46904  ichn  47430