Theorem con2bi 354

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 319	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜑 ↔ ¬ ¬ 𝜓))
2		notnotb 315	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3	2	bibi2i 338	. 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ ¬ 𝜓))
4		bicom 221	. 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
5	1, 3, 4	3bitr2i 299	1 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205

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 206

This theorem is referenced by:  con2bid  355  nbbn  385