Theorem xor 861

Description: Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)

Assertion

Ref	Expression
xor	⊢ (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))

Proof of Theorem xor

Step	Hyp	Ref	Expression
1		iman 413	. . . 4 ⊢ ((φ → ψ) ↔ ¬ (φ ∧ ¬ ψ))
2		iman 413	. . . 4 ⊢ ((ψ → φ) ↔ ¬ (ψ ∧ ¬ φ))
3	1, 2	anbi12i 678	. . 3 ⊢ (((φ → ψ) ∧ (ψ → φ)) ↔ (¬ (φ ∧ ¬ ψ) ∧ ¬ (ψ ∧ ¬ φ)))
4		dfbi2 609	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
5		ioran 476	. . 3 ⊢ (¬ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)) ↔ (¬ (φ ∧ ¬ ψ) ∧ ¬ (ψ ∧ ¬ φ)))
6	3, 4, 5	3bitr4ri 269	. 2 ⊢ (¬ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)) ↔ (φ ↔ ψ))
7	6	con1bii 321	1 ⊢ (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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  df-or 359  df-an 360

This theorem is referenced by:  dfbi3  863  pm5.24  864  4exmid  905  excxor  1309  elsymdif  3223  symdif2  3520