Theorem nbi2 1028

Description: Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)

Assertion

Ref	Expression
nbi2	⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Proof of Theorem nbi2

Step	Hyp	Ref	Expression
1		xor3 384	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
2		pm5.17 1024	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))
3	1, 2	bitr4i 280	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858

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 209  df-an 400  df-or 859

This theorem is referenced by:  xor2  1536