Theorem notbi 322

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 321	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3		id 22	. . 3 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	con4bid 320	. 2 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑 ↔ 𝜓))
5	2, 4	impbii 212	1 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 209

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 210

This theorem is referenced by:  notbii  323  con4bii  324  con2bi  357  nbn2  374  pm5.32  577  norass  1539  hadnot  1609  had0  1611  cbvexdw  2340  cbvexd  2407  rexbi  3154  vtoclgft  3458  rexprg  4598  isocnv3  7119  suppimacnv  7894  sumodd  15912  f1omvdco3  18795  ist0cld  31451  onsuct0  34316  bj-cbvexdv  34668  wl-3xornot  35338  ifpbi1  40710  ifpbi13  40722  abciffcbatnabciffncba  44039  abciffcbatnabciffncbai  44040  ichn  44524