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

Step	Hyp	Ref	Expression
1		bicom 191	. 2 ⊢ ((φ ↔ ¬ ψ) ↔ (¬ ψ ↔ φ))
2		dfbi2 609	. 2 ⊢ ((¬ ψ ↔ φ) ↔ ((¬ ψ → φ) ∧ (φ → ¬ ψ)))
3		orcom 376	. . . 4 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
4		df-or 359	. . . 4 ⊢ ((ψ ∨ φ) ↔ (¬ ψ → φ))
5	3, 4	bitr2i 241	. . 3 ⊢ ((¬ ψ → φ) ↔ (φ ∨ ψ))
6		imnan 411	. . 3 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))
7	5, 6	anbi12i 678	. 2 ⊢ (((¬ ψ → φ) ∧ (φ → ¬ ψ)) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))
8	1, 2, 7	3bitrri 263	1 ⊢ (((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)) ↔ (φ ↔ ¬ ψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  nbi2  862