Theorem xor2 1310

Description: Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
xor2	⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))

Proof of Theorem xor2

Step	Hyp	Ref	Expression
1		df-xor 1305	. 2 ⊢ ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))
2		nbi2 862	. 2 ⊢ (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))
3	1, 2	bitri 240	1 ⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ⊻ wxo 1304

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  df-xor 1305

This theorem is referenced by:  cador  1391  cad1  1398