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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
StepHypRef Expression
1 notbi 286 . 2 ((φ ↔ ¬ ψ) ↔ (¬ φ ↔ ¬ ¬ ψ))
2 notnot 282 . . 3 (ψ ↔ ¬ ¬ ψ)
32bibi2i 304 . 2 ((¬ φψ) ↔ (¬ φ ↔ ¬ ¬ ψ))
4 bicom 191 . 2 ((¬ φψ) ↔ (ψ ↔ ¬ φ))
51, 3, 43bitr2i 264 1 ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))
Colors of variables: wff setvar class
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
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