Theorem xor 1016

Description: Two ways to express exclusive disjunction (df-xor 1512). 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 401	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
2		iman 401	. . . 4 ⊢ ((𝜓 → 𝜑) ↔ ¬ (𝜓 ∧ ¬ 𝜑))
3	1, 2	anbi12i 628	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑)))
4		dfbi2 474	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
5		ioran 985	. . 3 ⊢ (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑)))
6	3, 4, 5	3bitr4ri 304	. 2 ⊢ (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (𝜑 ↔ 𝜓))
7	6	con1bii 356	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  pm5.24  1050  excxor  1516  elsymdif  4238  rpnnen2lem12  16248  ist0-3  23288  eliuniincex  45100  eliincex  45101  abnotataxb  46912  ldepslinc  48452