Theorem nbi2 862

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 346	. 2 ⊢ (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))
2		pm5.17 858	. 2 ⊢ (((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)) ↔ (φ ↔ ¬ ψ))
3	1, 2	bitr4i 243	1 ⊢ (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ 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:  xor2  1310