Theorem notbi 311

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 310	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3		id 22	. . 3 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	con4bid 309	. 2 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑 ↔ 𝜓))
5	2, 4	impbii 201	1 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 198

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 199

This theorem is referenced by:  notbii  312  con4bii  313  con2bi  346  nbn2  363  pm5.32  566  hadnot  1565  had0  1567  cbvexd  2342  isocnv3  6908  suppimacnv  7644  sumodd  15599  f1omvdco3  18338  onsuct0  33315  bj-cbvexdv  33587  ifpbi1  39245  ifpbi13  39257  abciffcbatnabciffncba  42601  abciffcbatnabciffncbai  42602  ichn  43002