Theorem con2bi 318

Description: Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Assertion

Ref	Expression
con2bi	⊢ ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))

Proof of Theorem con2bi

Step	Hyp	Ref	Expression
1		notbi 286	. 2 ⊢ ((φ ↔ ¬ ψ) ↔ (¬ φ ↔ ¬ ¬ ψ))
2		notnot 282	. . 3 ⊢ (ψ ↔ ¬ ¬ ψ)
3	2	bibi2i 304	. 2 ⊢ ((¬ φ ↔ ψ) ↔ (¬ φ ↔ ¬ ¬ ψ))
4		bicom 191	. 2 ⊢ ((¬ φ ↔ ψ) ↔ (ψ ↔ ¬ φ))
5	1, 3, 4	3bitr2i 264	1 ⊢ ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))

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:  con2bid  319  nbbn  347