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Theorem pm5.17 858
 Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
Assertion
Ref Expression
pm5.17 (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ))

Proof of Theorem pm5.17
StepHypRef Expression
1 bicom 191 . 2 ((φ ↔ ¬ ψ) ↔ (¬ ψφ))
2 dfbi2 609 . 2 ((¬ ψφ) ↔ ((¬ ψφ) (φ → ¬ ψ)))
3 orcom 376 . . . 4 ((φ ψ) ↔ (ψ φ))
4 df-or 359 . . . 4 ((ψ φ) ↔ (¬ ψφ))
53, 4bitr2i 241 . . 3 ((¬ ψφ) ↔ (φ ψ))
6 imnan 411 . . 3 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
75, 6anbi12i 678 . 2 (((¬ ψφ) (φ → ¬ ψ)) ↔ ((φ ψ) ¬ (φ ψ)))
81, 2, 73bitrri 263 1 (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  nbi2  862
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