Theorem xor2 1510

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 1505	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		nbi2 1012	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
3	1, 2	bitri 275	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 844   ⊻ wxo 1504

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 206  df-an 396  df-or 845  df-xor 1505

This theorem is referenced by:  xoror  1511  xornan  1512  cador  1601  saddisjlem  16404  wl-df4-3mintru2  36859  ifpdfxor  42752  dfxor4  43031  nanorxor  43578