Theorem notbi 286

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 19	. . 3 ⊢ ((φ ↔ ψ) → (φ ↔ ψ))
2	1	notbid 285	. 2 ⊢ ((φ ↔ ψ) → (¬ φ ↔ ¬ ψ))
3		id 19	. . 3 ⊢ ((¬ φ ↔ ¬ ψ) → (¬ φ ↔ ¬ ψ))
4	3	con4bid 284	. 2 ⊢ ((¬ φ ↔ ¬ ψ) → (φ ↔ ψ))
5	2, 4	impbii 180	1 ⊢ ((φ ↔ ψ) ↔ (¬ φ ↔ ¬ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176

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 177

This theorem is referenced by:  notbii  287  con4bii  288  con2bi  318  nbn2  334  pm5.32  617  cbvexd  2009  rexxpf  4829